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%!TEX root = forallxyyc.tex
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\begin_layout Part
Dedução natural para LVF
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name "ch.NDTFL"

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\begin_layout Chapter
A ideia de dedução natural
\end_layout

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name "s:NDVeryIdea"

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\begin_layout Standard
Voltando a 
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S
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plural "false"
caps "false"
noprefix "false"

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, dissemos que um argumento é válido sse não há caso no qual todas as premissas
 são verdadeiras e a conclusão é falsa.
 
\end_layout

\begin_layout Standard
No caso de LVF, isto nos levou a desenvolver as tabelas de verdade.
 Cada linha da tabela de verdade completa corresponde a uma valoração.
 Assim, quando confrontados com um argumento fr LVF, temos uma forma bastante
 direta de avaliar se há uma valoração na qual as premissas são verdadeiras
 e a conclusão é falsa: use a tabela de verdade.
 Entretanto, as tabelas de verdade não nos dão necessariamente muitos 
\emph on
insight
\emph default
.
 Considere dois argumentos em LVF:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align*}
P\eor Q,\enot P & \therefore Q\\
P\eif Q,P & \therefore Q
\end{align*}

\end_inset

Claramente, estes são argumentos válidos.
 Você pode confirmar que eles são válidos, contruindo tabelas de verdade
 com quatro linhas, mas poderíamos dizer que eles usam diferentes 
\emph on
formas
\emph default
 de raciocínio.
 Poderia ser interessante levar em conta estas formas diferentes de inferância.
 
\end_layout

\begin_layout Standard
Um objetivo de um 
\emph on
sistema de dedução natural
\emph default
 é mostrar que argumentos particulares são válidos, de forma que nos ajude
 a entender o raciocínio que os argumentos poderiam envolver.
 Começamos com regras bastante básicas de inferência.
 Estas regras podem ser combinadas para oferecer argumentos mais complicados.
 De fato, com apenas um pequeno pacote inicial de inferência, esperamos
 capturar todos os argumentos válidos.
\end_layout

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\emph on
Isto é uma forma muito diferente de pensar sobre argumentos.

\emph default
 
\end_layout

\begin_layout Standard
Com as tabelas de verdade, consideramos diretamente formas distintas para
 tornar sentenças verdadeiras ou falsas.
 Com sistemas de dedução natural, manipulamos sentenças de acordo com as
 regras que estabelecemos como boas regras.
 Sistema de dedução natural promete-nos dar um melhor 
\emph on
insight
\emph default
 — ou pelo menos, um 
\emph on
insight
\emph default
 diferente — sobre como os argumentos funcionam.
 
\end_layout

\begin_layout Standard
A mudança para dedução natural poderia ser motivada por mais do que a busca
 por 
\emph on
insight
\emph default
.
 Ela poderia também ser motivada por 
\emph on
necessidade
\emph default
.
 Considere: 
\begin_inset Formula 
\[
A_{1}\eif C_{1}\therefore(A_{1}\eand A_{2}\eand A_{3}\eand A_{4}\eand A_{5})\eif(C_{1}\eor C_{2}\eor C_{3}\eor C_{4}\eor C_{5})
\]

\end_inset

Para testar a validade deste argumento, poderíamos usar uma tabela de verdade
 com 1024 linhas.
 Se você fizer isso corretamente, então você verá que não há linha na qual
 todas as premissas são verdadeiras e na qual a conclusão é falsa.
 Assim, você saberá que o argumento é válido (mas, como justamente mencionado,
 há um sentido no qual você não saberá 
\emph on
por que
\emph default
 o argumento é válido).
 Mas agora considere: 
\begin_inset Formula 
\begin{align*}
A_{1}\eif C_{1}\therefore\  & (A_{1}\eand A_{2}\eand A_{3}\eand A_{4}\eand A_{5}\eand A_{6}\eand A_{7}\eand A_{8}\eand A_{9}\eand A_{10})\eif\phantom{(}\\
 & (C_{1}\eor C_{2}\eor C_{3}\eor C_{4}\eor C_{5}\eor C_{6}\eor C_{7}\eor C_{8}\eor C_{9}\eor C_{10})
\end{align*}

\end_inset

Este argumento também é válido — como você provavelmente dirá —, mas para
 testar isso é requerido uma tabela de verdade com 
\begin_inset Formula $2^{20}=1048576$
\end_inset

 linhas.
 Em tese, podemos configurar uma máquina para analisar as tabelas de verdade
 e informar quando terminar.
 Na prática, argumentos complicados em LVF podem tornar-se 
\emph on
intratáveis
\emph default
, se usarmos tabelas de verdade.
 
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\begin_layout Standard
Não obstante, quando chegarmos à lógica de primeira ordem (LPO) (começando
 no capítulo 
\begin_inset CommandInset ref
LatexCommand ref
reference "s:FOLBuildingBlocks"
plural "false"
caps "false"
noprefix "false"

\end_inset

), o problema ficará dramaticamente pior.
 Não há nada como teste de tabela de verdade para (LPO).
 Para avaliar se um argumento é válido ou não, temos de raciocinar sobre
 
\emph on
todas
\emph default
 as interpretações, mas, como veremos, há infinitas interretações possíveis.Nem
 mesmo em tese podemos configurar uma máquina para analisar as infinitas
 interpretações possíveis e informar quando terminar: ela 
\emph on
nunca
\emph default
 terminará.
 Precisamos criar uma maneira mais eficiente de raciocinar sobre todas as
 interpretações ou precisamos procurar algo diferente.
 
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\begin_layout Standard
De fato, há sistemas que codificam maneiras de raciocinar sobre todas as
 interpretações.
 Eles foram desenvolvidos nos anos 1950 por Evert Beth e Jaakko Hintikka,
 mas não seguiremos este caminho.
 Em vez disso, olharemos para dedução natural.
\end_layout

\begin_layout Standard
Em vez de raciocinar diretamente sobre todas as valorações (no caso de LVF),
 tentaremos selecionar algumas regras básicas de inferência.
 Algumas delas governarão o comportamento dos conectivos sentenciais.
 Outras governarão o comportamento dos quantificadores e da identidade que
 são marcas registradas de LPO.
 O sistema de regras resultante dar-nos-ão uma nova maneira de pensar sobre
 a validade de argumentos.
 O desenvolvimento moderno de dedução natural é datado a partir de artigos
 publicados simultanea e independentemente por Getzen Gerhard and Stanisław
 Jaśkowski (1934).
 Todavia, o sistema de dedução natural que iremos considerar é amplamente
 baseado em torno do trabalho de Frederic Fitch (publicado primeiro em 1952).
\end_layout

\begin_layout Chapter
Regras básicas para LVF
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name "s:BasicTFL"

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Desenvolveremos um sistema de 
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dedu
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c{c}
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~a
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o natural
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}
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.
 Para cada conectivo, haverá regras de 
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introdu
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~a
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o
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, que nos permite provar uma sentença que tem este conectivo como o operador
 lógico principal e regras de 
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elimina
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~a
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o
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, que nos permite provar algo dada uma sentença que tem este conectivo como
 operador lógico principal.
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A ideia de uma prova formal
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Uma 
\emph on
prova formal
\emph default
 é uma sequência de sentenças, algumas das quais são marcadas como sendo
 suposições iniciais (ou premissas).
 A última linha da prova formal é a conclusão (doravante, chamaremos simplesment
e estas rovas', mas você deveria estar consciente que há também 
\emph on
provas informais
\emph default
).
 Como uma ilustração, considere:
\begin_inset Formula 
\[
\enot(A\eor B)\therefore\enot A\eand\enot B
\]

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Começaremos uma prova, escrevendo a premissa:
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\backslash
begin{proof} 
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enot (A 
\backslash
eor B)} 
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\backslash
end{proof}
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\end_layout

\begin_layout Standard
Perceba que numeramos a premissa, uma vez que queremos referir-nos à premissa
 novamente.
 De fato, qualquer linha na prova é numerada, de forma que podemos referir-nos
 novamente.
 
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\begin_layout Standard
Note também que traçamos uma linha embaixo da premissa.
 Tudo escrito acima da linha é uma 
\emph on
suposição
\emph default
.
 Qualquer coisa escrita abaixo da linha será algo que se segue das suposições
 ou será alguma nova suposição.
 Esperamos concluir `
\begin_inset Formula $\enot A\eand\enot B$
\end_inset

'; desse modo, esperamos, enfim, concluir nossa prova com
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\backslash
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\backslash
have[n]{con}{
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\backslash
eand 
\backslash
enot B} 
\end_layout

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\backslash
end{proof}
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\begin_layout Standard
\noindent
para algum número 
\begin_inset Formula $n$
\end_inset

.
 Não importa em qual número de linhas terminamos, mas, obviamente, preferiríamos
 uma prova curta a uma longa.
\end_layout

\begin_layout Standard
Similarmente, suponha que desejássemos considerar:
\begin_inset Formula 
\[
A\eor B,\enot(A\eand C),\enot(B\eand\enot D)\therefore\enot C\eor D
\]

\end_inset

O argumento tem três premissas, assim começamos escrevendo todas, numerando-as
 e desenhando um linha debaixo delas: 
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\backslash
begin{proof} 
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\backslash
hypo{a1}{A 
\backslash
eor B} 	
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\begin_layout Plain Layout


\backslash
hypo{a2}{
\backslash
enot (A
\backslash
eand C)} 	
\end_layout

\begin_layout Plain Layout


\backslash
hypo{a3}{
\backslash
enot (B 
\backslash
eand 
\backslash
enot D)} 
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\backslash
end{proof}
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\noindent
e esperamos concluir com alguma linha:
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\backslash
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\backslash
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\backslash
end{proof}
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\begin_layout Standard
\noindent
Tudo que resta fazer é explicar cada uma das regras que podemos usar ao
 longo do percurso das premissas à conclusão.
 As regras são separadas pelos nossos conectivos lógicos.
\end_layout

\begin_layout Section
Reiteração
\end_layout

\begin_layout Standard
A primeira regra é tão incrivelmente óbvia que é surpreendente que nos importamo
s com ela.
 
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%The very first rule is so breathtakingly obvious that it is surprising
 we bother with it at all.
 
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%If you already have shown something in the course of a proof, the 
\backslash
emph{reiteration rule} allows you to repeat it on a new line.
 For example:
\end_layout

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\end_layout

\end_inset

Se você já mostrou algo no curso de uma prova, a 
\emph on
regra de reiteração
\emph default
 permite que você repita-o em uma nova linha.
 Por exemplo: 
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4
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A 
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 B
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R
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%This indicates that we have written `$A 
\backslash
eand B$' on line~$4$.
 Now, at some later line---line~$10$, for example---we have decided that
 we want to repeat this.
 So we write it down again.
 We also add a citation which justifies what we have written.
 In this case, we write `R', to indicate that we are using the reiteration
 rule, and we write `$4$', to indicate that we have applied it to line $4$.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Isto indica que escrevemos `
\begin_inset Formula $A\eand B$
\end_inset

' na line
\begin_inset space ~
\end_inset


\begin_inset Formula $4$
\end_inset

.
 Agora, em uma linha posterior, — linha
\begin_inset space ~
\end_inset


\begin_inset Formula $10$
\end_inset

, por exemplo —, decidimos que queremos repetir isto.
 Assim, escrevemo-la novamente.
 Também adicionamos uma citação que justifica o que escrevemos.
 Neste caso, escrevemos `R' para indicar que estamos usando a regra de reiteraçã
o e escrevemos `
\begin_inset Formula $4$
\end_inset

' para indicar a regra foi aplicada à linha `
\begin_inset Formula $4$
\end_inset

'.
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\begin_layout Standard
Aqui está uma expressão geral da regra:
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%Here is a general expression of the rule:
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\begin_layout Plain Layout

\end_layout

\end_inset


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\backslash
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\backslash
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m
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R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The point is that, if any sentence $
\backslash
meta{A}$ occurs on some line, then we can repeat $
\backslash
meta{A}$ on later lines.
 Each line of our proof must be justified by some rule, and here we have
 `R $m$'.
 This means: Reiteration, applied to line~$m$.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

O ponto é que, se qualquer sentença 
\begin_inset Formula $\meta{A}$
\end_inset

 ocorre em alguma linha, então podemos repetir 
\begin_inset Formula $\meta{A}$
\end_inset

 em linhas posteriores.
 Cada linha de nossa prova deve ser justificada por alguma regra e aqui
 temos `R 
\begin_inset Formula $m$
\end_inset

'.
 Isto significa: Reiteração, aplicada à linha 
\begin_inset space ~
\end_inset


\begin_inset Formula $m$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Two things need emphasizing.
 First `$
\backslash
meta{A}$' is not a sentence of TFL.
 Rather, it a symbol in the metalanguage, which we use when we want to talk
 about any sentence of TFL (see 
\backslash
S
\backslash
ref{s:UseMention}).
 Second, and similarly, `$m$' is not a symbol that will appear on a proof.
 Rather, it is a symbol in the metalanguage, which we use when we want to
 talk about any line number of a proof.
 In an actual proof, the lines are numbered `$1$', `$2$', `$3$', and so
 forth.
 But when we define the rule, we use variables like `$m$' to underscore
 the point that the rule may be applied at any point.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Duas coisas precisam ser enfatizadas.
 Em primeiro lugar, `
\begin_inset Formula $\meta{A}$
\end_inset

' não é uma sentença de LVF.
 Em vez disso, ele é um símbolo na metalinguagem, que usamos quando queremos
 falar sobre qualquer sentença de LVF (veja 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:UseMention"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
 Em segundo lugar, da mesma forma, `
\begin_inset Formula $m$
\end_inset

' não é um símbolo que aparecerá em uma prova.
 Em vez disso, ele é um símbolo na metalinguagem, que usamos quando queremos
 falar sobre qualquer linha de uma prova.
 Em uma prova real, as linhas são numeradas por `
\begin_inset Formula $1$
\end_inset

', `
\begin_inset Formula $2$
\end_inset

', `
\begin_inset Formula $3$
\end_inset

' e assim por diante.
 Mas quando definimos a regra, usamos variáveis como `
\begin_inset Formula $m$
\end_inset

' para sublinhar o ponto ao qual a regra pode ser aplicada em qualquer momento.
\end_layout

\begin_layout Section
Conjunção
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Suppose we want to show that Ludwig is both reactionary and libertarian.
 One obvious way to do this would be as follows: first we show that Ludwig
 is reactionary; then we show that Ludwig is libertarian; then we put these
 two demonstrations together, to obtain the conjunction.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Suponha que desejamos provar que Ludwig é tanto reacionário como libertário.
 Uma maneira óbvia de fazer isto seria como se segue: primeiro, mostre que
 Ludwig é reacionário; então mostre que Ludwig é libertário; depois coloque
 estas duas demonstrações juntas para obter a conjunção.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Our natural deduction system will capture this thought straightforwardly.
 In the example given, we might adopt the following symbolization key:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Nosso sistema de dedução natural capturará este pensamento diretamente.
 No exemplo dado, poderíamos adotar a seguinte chave de simbolização: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{ekey}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item
\end_layout

\end_inset

[R] Ludwig é reacionário
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%is reactionary
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item
\end_layout

\end_inset

[L] Ludwig é libertário
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%is libertarian
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{ekey}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Perhaps we are working through a proof, and we have obtained `$R$' on line
 8 and `$L$' on line 15.
  Then on any subsequent line we can obtain `$R 
\backslash
eand L$' thus:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Talvez estejamos trabalhando através de uma prova e obtivemos `
\begin_inset Formula $R$
\end_inset

' na linha 8 e `
\begin_inset Formula $L$
\end_inset

' na linha 15.
 Então, na linha subsequente, podemos obter `
\begin_inset Formula $R\eand L$
\end_inset

' desse modo: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

15
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a, b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Note that every line of our proof must either be an assumption, or must
 be justified by some rule.
 We cite `$
\backslash
eand$I 8, 15' here to indicate that the line is obtained by the rule of
 conjunction introduction ($
\backslash
eand$I) applied to lines 8 and 15.
 We could equally well obtain:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Note que qualquer linha de nossa prova deve ser uma suposição ou deve ser
 justificada por alguma regra.
 Citamos `
\begin_inset Formula $\eand$
\end_inset

I 8, 15' aqui para indicar que a linha é obtida pela regra de introdução
 da conjunção (
\begin_inset Formula $\eand$
\end_inset

I) aplicada às linhas 8 e 15.
 Da mesma forma, podedríamos obter também: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

15
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

L 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b, a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%with the citation reversed, to reflect the order of the conjuncts.
 More generally, here is our conjunction introduction rule:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

com a citação invertida para refletir a ordem dos conjunctos.
 Falando de forma mais geral, aqui está nossa regra de introdução: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a, b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%To be clear, the statement of the rule is 
\backslash
emph{schematic}.
 It is not itself a proof.
  `$
\backslash
meta{A}$' and `$
\backslash
meta{B}$' are not sentences of TFL.
 Rather, they are symbols in the metalanguage, which we use when we want
 to talk about any sentence of TFL (see 
\backslash
S
\backslash
ref{s:UseMention}).
 Similarly, `$m$' and `$n$' are not a numerals that will appear on any actual
 proof.
 Rather, they are symbols in the metalanguage, which we use when we want
 to talk about any line number of any proof.
 In an actual proof, the lines are numbered `$1$', `$2$', `$3$', and so
 forth, but when we define the rule, we use variables to emphasize that
 the rule may be applied at any point.
  The rule requires only that we have both conjuncts available to us somewhere
 in the proof.
 They can be separated from one another, and they can appear in any order.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Esclarecendo, o enunciado da regra é 
\emph on
esquemática
\emph default
.
 Ele não é uma prova em si.
 `
\begin_inset Formula $\meta{A}$
\end_inset

' e `
\begin_inset Formula $\meta{B}$
\end_inset

' não são sentenças de LVF.
 Em vez disso, eles são símbolos na metalinguagem, que usamos quando queremos
 falar sobre qualquer sentença de LVF (veja 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:UseMention"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
 Similarmente, `
\begin_inset Formula $m$
\end_inset

' e `
\begin_inset Formula $n$
\end_inset

' não são numerais que aparecem em qualquer prova real.
 Em vez disso, eles são símbolos na metalinguagem, que usamos quando queremos
 falar sobre qualquer linha de qualquer prova.
 Na prova real, as linhas são numeradas `
\begin_inset Formula $1$
\end_inset

', `
\begin_inset Formula $2$
\end_inset

', `
\begin_inset Formula $3$
\end_inset

' e assim por diante, mas quando definimos a regra, usamos variáveis para
 enfatizar que a regra pode ser aplicada a qualquer momento.
 A regra requer apenas que tenhamos ambos conjunctos disponíveis em algum
 lugar na prova.
 Eles podem ser separados um do outro e eles podem aparecer em qualquer
 ordem.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The rule is called `conjunction 
\backslash
emph{introduction}' because it introduces the symbol `$
\backslash
eand$' into our proof where it may have been absent.
 Correspondingly, we have a rule that 
\backslash
emph{eliminates} that symbol.
  Suppose you have shown that Ludwig is both reactionary and libertarian.
 You are entitled to conclude that Ludwig is reactionary.
 Equally, you are entitled to conclude that Ludwig is libertarian.
 Putting this together, we obtain our conjunction elimination rule(s):
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A regra é chamada 
\emph on
introdução
\emph default
 da conjunção, porque ela introduz o símbolo `
\begin_inset Formula $\eand$
\end_inset

' na nossa prova onde ele poderia estar ausente.
 De forma semelhante, temos a regra que 
\emph on
elimina
\emph default
 este símbolo.
 Suponha que mostramos que Ludwig é tanto reacionário como libertário.
 Você tem o direito [
\emph on
You are entitled to
\emph default
] de concluir que Ludwig é reacionário.
 Da mesma forma, você tem o direito de concluir que Ludwig é libertário.
 Agrupando isto, obtemos nossa(s) regra(s) de eliminação da conjunção: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e também: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\noindent
O ponto é simplesmente que, quando você tem uma conjunção em alguma linha
 de uma prova, você pode obter um dos conjunctos por 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

E.
 Um ponto que vale a pena enfatizar: você só pode aplicar esta regra quando
 a conjunção é o operador lógico principal.
 Desse modo, você não pode inferir `
\begin_inset Formula $D$
\end_inset

' a partir de `
\begin_inset Formula $C\eor(D\eand E)$
\end_inset

'!
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Even with just these two rules, we can start to see some of the power of
 our formal proof system.
 Consider:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Mesmo com apenas estas duas regras, podemos começar a ver algum poder de
 nosso sistema formal de prova.
 Considere: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item
\end_layout

\end_inset

[] 
\begin_inset Formula $[(A\eor B)\eif(C\eor D)]\eand[(E\eor F)\eif(G\eor H)]$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item
\end_layout

\end_inset

[
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
therefore
\end_layout

\end_inset

] 
\begin_inset Formula $[(E\eor F)\eif(G\eor H)]\eand[(A\eor B)\eif(C\eor D)]$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The main logical operator in both the premise and conclusion of this argument
 is `$
\backslash
eand$'.
 In order to provide a proof, we begin by writing down the premise, which
 is our assumption.
 We draw a line below this: everything after this line must follow from
 our assumptions by (repeated applications of) our rules of inference.
 So the beginning of the proof looks like this:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

O operador lógico principal tanto na premissa como na conclusão deste argumento
 é `
\begin_inset Formula $\eand$
\end_inset

'.
 A fim de fornecer uma prova, começamos escrevendo a premissa, que é nossa
 suposição.
 Traçamos uma linha debaixo dela: tudo depois desta linha deve seguir-se
 das nossas suposições por meio (de aplicações reiteradas de) nossas regras
 de inferência.
 Assim, o início da prova tem o seguinte aspecto: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

[(A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 B)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

(C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 D)] 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 [(E 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 F) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (G
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 H)]
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%From the premise, we can get each of the conjuncts by {
\backslash
eand}E.
 The proof now looks like this:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A partir da premissa, podemos obter vada um dos conjunctos, aplicando 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

E.
 A prova tem o seguinte aspecto: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

[(A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 B)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

(C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 D)] 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 [(E 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 F) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (G
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 H)]
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

[(A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 B)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

(C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 D)]
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

[(E 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 F) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (G
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 H)]
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 So by applying the 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

I rule to lines 3 and 2 (in that order), we arrive at the desired conclusion.
 The finished proof looks like this: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

[(A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 B)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

(C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 D)] 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 [(E 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 F) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (G
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 H)]
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
par 
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

[(A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 B)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

(C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 D)]
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

[(E 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 F) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (G
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 H)]
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ba
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

[(E 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 F) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (G
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 H)] 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 [(A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 B)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

(C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 D)]
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b,a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This is a very simple proof, but it shows how we can chain rules of proof
 together into longer proofs.
 In passing, note that investigating this argument with a truth table would
 have required 256 lines; our formal proof required only four lines.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Isto é uma prova muito simples, mas ela mostra como podemos encadear as
 regras de prova em provas mais longas.
 A propósito [
\emph on
in passing
\emph default
], observe que investigar este argumento usando tabela de verdade teria
 exigido 256 linhas; nossa prova formal exigiu 4 linhas.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%It is worth giving another example.
 Back in 
\backslash
S
\backslash
ref{s:MoreBracketingConventions}, we noted that this argument is valid:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Vale a pena dar um outro exemplo.
 Voltando a 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:MoreBracketingConventions"
plural "false"
caps "false"
noprefix "false"

\end_inset

, notamos que este argumento é válido: 
\begin_inset Formula 
\[
A\eand(B\eand C)\therefore(A\eand B)\eand C
\]

\end_inset

Para dar uma prova que corresponda a este argumento, começamos escrevendo:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%To provide a proof corresponding to this argument, we start by writing:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 A partir da premissa, podemos obter cada um dos conjunctos, aplicando 
\begin_inset Formula $\eand$
\end_inset

E duas vezes.
 Podemos, então, aplicar 
\begin_inset Formula $\eand$
\end_inset

E mais duas vezes, assim nossa prova é: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%From the premise, we can get each of the conjuncts by applying $
\backslash
eand$E twice.
 We can then apply $
\backslash
eand$E twice more, so our proof looks like:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 Mas, felizmente, agora podemos reintroduzir conjunções na ordem que desejamos,
 de forma que nossa prova final é: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%But now we can merrily reintroduce conjunctions in the order we wanted
 them, so that our final proof is:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

abc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

abc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

abc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a, b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab, c
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Recall that our official definition of sentences in TFL only allowed conjunctio
ns with two conjuncts.
 The proof just given suggests that we could drop inner brackets in all
 of our proofs.
 However, this is not standard, and we will not do this.
 Instead, we will maintain our more austere bracketing conventions.
 (Though we will still allow ourselves to drop outermost brackets, for legibilit
y.)
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Lembre-se que nossa definição oficial de sentenças em LVF apenas permitia
 conjunções com dois conjunctos.
 A prova justamante dada sugere que poderíamos excluir os parênteses internos
 em todas nossas provas.
 Entretanto, isto não é padrão.
 Em vez disso, manteremos nossas convenções mais austeras sobre parênteses
 (embora ainda permitiremos a exclusão dos parênteses mais externos, o que
 torna as coisas mais legível).
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Daremos uma ilustração final.
 Ao usar a regra 
\begin_inset Formula $\eand$
\end_inset

I, não há qualquer exigência para aplicá-la a diferentes sentenças.
 Portanto, se quisermos, podemos formalmente provar `
\begin_inset Formula $A\eand A$
\end_inset

' a partir de `
\begin_inset Formula $A$
\end_inset

' dessa forma: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Let's give one final illustration.
 When using the $
\backslash
eand$I rule, there is no requirement to apply it to different sentences.
 So, if we want, we can formally prove `$A 
\backslash
eand A$' from `$A$' thus:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

aa
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a, a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 Simples, mas efetivo.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Simple, but effective.
\end_layout

\end_inset


\end_layout

\begin_layout Section
Condicional
\end_layout

\begin_layout Standard
Considere o seguinte argumento:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Consider the following argument:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item
\end_layout

\end_inset

[] Se Jane é esperta, então ela é rápida.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%If Jane is smart then she is fast.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item
\end_layout

\end_inset

[] Jane é esperta.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Jane is smart.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item
\end_layout

\end_inset

[
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
therefore
\end_layout

\end_inset

] Jane é rápida.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Jane is fast.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 Certamente, este argumento é válido e sugere diretamente uma regra de eliminaçã
o do condicional (
\begin_inset Formula $\eif$
\end_inset

E):
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This argument is certainly valid, and it suggests a straightforward conditional
 elimination rule ($
\backslash
eif$E):
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ce{ab,a}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This rule is also sometimes called 
\backslash
emph{modus ponens}.
 Again, this is an elimination rule, because it allows us to obtain a sentence
 that may not contain `$
\backslash
eif$', having started with a sentence that did contain `$
\backslash
eif$'.
  Note that the conditional $
\backslash
meta{A}
\backslash
eif
\backslash
meta{B}$ and the antecedent~$
\backslash
meta{A}$ can be separated from one another in the proof, and they can appear
 in any order.
 However, in the citation for $
\backslash
eif$E, we always cite the conditional first, followed by the antecedent.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Esta regra é também 
\emph on
modus ponens
\emph default
.
 Novamente, isto é uma regra de eliminação, porque ela permite a obtenção
 de uma sentença que pode não conter `
\begin_inset Formula $\eif$
\end_inset

', depois de iniciarmos com uma sentença que continha `
\begin_inset Formula $\eif$
\end_inset

' como operador lógico principal.
 Note que o condicional 
\begin_inset Formula $\meta{A}\eif\meta{B}$
\end_inset

 e o antecedente
\begin_inset space ~
\end_inset


\begin_inset Formula $\meta{A}$
\end_inset

 podem ser separados um do outro na prova e eles podem ocorrer em qualquer
 ordem.
 Todavia, na citação para 
\begin_inset Formula $\eif$
\end_inset

E, sempre citamos o condicional primeiro, seguido pelo antecedente.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The rule for conditional introduction is also quite easy to motivate.
 The following argument should be valid:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A regra para introdução do condicional é também muito fácil de dar uma motivação.
 O seguinte argumento deveria ser válido: 
\end_layout

\begin_layout Quote
Ludwig é reacionário.
 Portanto, se Ludwig é libertário, então Ludwig é reacionário 
\emph on
e
\emph default
 libertário.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Ludwig is reactionary.
 Therefore if Ludwig is libertarian, then Ludwig is both reactionary 
\backslash
emph{and} libertarian.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
Se alguém duvidasse que isto era válido, poderíamos tentar convencê-lo do
 contrário, explicando da seguinte forma: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%If someone doubted that this was valid, we might try to convince them otherwise
 by explaining ourselves as follows:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Quote
Assuma que Ludwig é reacionário.
 Ora, 
\emph on
além disso
\emph default
 assuma que Ludwig é libertário.
 Então, pela introdução da conjunção — que acabamos de discutir —, Ludwig
 é reacionário e libertário.
 É calro, isto é um condicional na suposição de que Ludwig é libertário.
 Mas isto justamente significa que se Ludwig é libertário, então ele é reacionár
io e libertário.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Assume that Ludwig is reactionary.
 Now, 
\backslash
emph{additionally} assume that Ludwig is libertarian.
 Then by conjunction introduction---which we just discussed---Ludwig is
 both reactionary and libertarian.
 Of course, that's conditional on the assumption that Ludwig is libertarian.
 But this just means that, if Ludwig is libertarian, then he is both reactionary
 and libertarian.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
Transferindo para o formato de dedução natural, aqui está o padrão de raciocínio
 que acabamos de usar.
 Começamos com uma premissa, `Ludwig é reacionário', assim: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Transferred into natural deduction format, here is the pattern of reasoning
 that we just used.
 We started with one premise, `Ludwig is reactionary', thus:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

r
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The next thing we did is to make an 
\backslash
emph{additional} assumption (`Ludwig is libertarian'), for the sake of argument.
 To indicate that we are no longer dealing 
\backslash
emph{merely} with our original assumption (`$R$'), but with some additional
 assumption, we continue our proof as follows:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A próxima coisa que fizemos é assumir uma suposição adicional (`Ludwig é
 libertário'), para o propósito do argumento.
 Para indicar que não estamos mais lidando 
\emph on
meramente
\emph default
 com a suposição inicial (`
\begin_inset Formula $R$
\end_inset

'), mas com alguma suposição adicional, continuamos a prova como se segue:
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

r
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Note that we are 
\backslash
emph{not} claiming, on line 2, to have proved `$L$' from line 1, so we do
 not write in any justification for the additional assumption on line 2.
 We do, however, need to mark that it is an additional assumption.
 We do this by drawing a line under it  (to indicate that it is an assumption)
 and by indenting it with a further vertical line (to indicate that it is
 additional).
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Note que não estamos reivindicando, na linha 2, ter provado `
\begin_inset Formula $L$
\end_inset

' a partir da linha 1, assim não escrevemos nesta linha qualquer justificação
 para a suposição adicional na linha 2.
 Todavia, precisamos marcar que ela é uma suposição adicional.
 Fazemos isto, traçando uma linha sob esta suposição (para indicar que ela
 é uma suposição) e recuando-a [
\emph on
indenting it
\emph default
] por meio da linha vertical (para indicar que ela é adicional).
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%	
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%With this extra assumption in place, we are in a position to use $
\backslash
eand$I.
 So we can continue our proof:
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Com esta suposição extra à disposição, estamos em posição de usar 
\begin_inset Formula $\eand$
\end_inset

I..
 Assim, podemos continuar nossa prova: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

r
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

rl
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

r, l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%			
\backslash
close
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%		
\backslash
have{con}{L 
\backslash
eif (R 
\backslash
eand L)}
\backslash
ci{l-rl}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%So we have now shown that, on the additional assumption, `$L$', we can
 obtain `$R 
\backslash
eand L$'.
 We can therefore conclude that, if `$L$' obtains, then so does `$R 
\backslash
eand L$'.
 Or, to put it more briefly, we can conclude `$L 
\backslash
eif (R 
\backslash
eand L)$':
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Desse modo, mostramos que, dada a suposição adicional, `
\begin_inset Formula $L$
\end_inset

', podemos obter `
\begin_inset Formula $R\eand L$
\end_inset

'.
 Portanto, podemos concluir que se `
\begin_inset Formula $L$
\end_inset

' ocorre, então `
\begin_inset Formula $R\eand L$
\end_inset

' ocorre.
 Ou, de forma mais breve, podemos concluir `
\begin_inset Formula $L\eif(R\eand L)$
\end_inset

': 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

r
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

rl
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

r, l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

L 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (R 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 L)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

l-rl
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Observe that we have dropped back to using one vertical line on the left.
 We have 
\backslash
emph{discharged} the additional assumption, `$L$', since the conditional
 itself follows just from our original assumption, `$R$'.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Observe que recuamos e voltamos a usar aquela linha vertical à esquerda.
 Nós 
\emph on
descartamos
\emph default
 a suposição adicional,`
\begin_inset Formula $L$
\end_inset

', uma vez que o próprio condicional se segue da nossa suposição original,
 `
\begin_inset Formula $R$
\end_inset

'.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The general pattern at work here is the following.
 We first make an additional assumption, $
\backslash
meta{A}$; and from that additional assumption, we prove~$
\backslash
meta{B}$.
 In that case, we know the following: If~$
\backslash
meta{A}$ is true, then~$
\backslash
meta{B}$ is true.
 This is wrapped up in the rule for conditional introduction:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

O padrão geral que está funcionando aqui é o seguinte.
 Em primeiro lugar, faça uma suposição adicional, 
\begin_inset Formula $\meta{A}$
\end_inset

; e desta suposição adicional, provamos
\begin_inset space ~
\end_inset


\begin_inset Formula $\meta{B}$
\end_inset

.
 Neste caso, sabemos o seguinte: Se
\begin_inset space ~
\end_inset


\begin_inset Formula $\meta{A}$
\end_inset

 é verdadeira, então
\begin_inset space ~
\end_inset


\begin_inset Formula $\meta{B}$
\end_inset

 é verdadeira.
 Isto está previsto na regra de introdução do condicional: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

i
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

j
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a-b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Podem existir muitas linhas ou poucas linhas entre as linhas 
\begin_inset Formula $i$
\end_inset

 e 
\begin_inset Formula $j$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% here can be as many or as few lines as you like between lines $i$ and
 $j$.
 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Será útil oferecer uma segunda ilustração mostrando o funcionamento de 
\begin_inset Formula $\eif$
\end_inset

I.
 Suponha que desejamos considerar o seguinte:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%It will help to offer a second  illustration of $
\backslash
eif$I in action.
 Suppose we want to consider the following:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset Formula 
\[
P\eif Q,Q\eif R\therefore P\eif R
\]

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We start by listing 
\backslash
emph{both} of our premises.
 Then, since we want to arrive at a conditional (namely, `$P 
\backslash
eif R$'), we additionally assume the antecedent to that conditional.
 Thus our main proof starts:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Começamos listando nossas premissas.
 Então, uma vez que queremos chegar a um condicional (a saber, `
\begin_inset Formula $P\eif R$
\end_inset

'), temos de assumir, além disso, o antecedente deste condicional.
 Portanto, nossa prova principal começa: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

pq
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 Q
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

qr
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Q 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

p
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Note that we have made `$P$' available, by treating it as an additional
 assumption, but now, we can use {
\backslash
eif}E on the first premise.
 This will yield `$Q$'.
 We can then use {
\backslash
eif}E on the second premise.
 So, by assuming `$P$' we were able to prove `$R$', so we apply the {
\backslash
eif}I rule---discharging `$P$'---and finish the proof.
 Putting all this together, we have: 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Note que temos `
\begin_inset Formula $P$
\end_inset

' disponível, quando a tratamos como uma suposição adicional, mas, agora,
 podemos usar 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

E na primeira premissa.
 Isto produzirá `
\begin_inset Formula $Q$
\end_inset

'.
 Podemos, então, usar 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

E na segunda premissa.
 Portanto, assumindo `
\begin_inset Formula $P$
\end_inset

', fomos capazes de provar `
\begin_inset Formula $R$
\end_inset

', logo aplicamos a regra 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

I — descartando `
\begin_inset Formula $P$
\end_inset

' — e terminamos a prova.
 Colocando tudo isto junto, temos: 
\begin_inset CommandInset label
LatexCommand label
name "HSproof"

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

pq
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 Q
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

qr
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Q 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

p
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

q
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Q
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ce{pq,p}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

r
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ce{qr,q}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

pr
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

p-r
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Suposições adicionais e subprovas
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Additional assumptions and subproofs
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A regra de 
\begin_inset Formula $\eif$
\end_inset

I invocou a ideia de fazer suposições adicionais.
 Estas suposições precisam ser manuseadas com algum cuidado.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The rule $
\backslash
eif$I invoked the idea of making additional assumptions.
 These need to be handled with some care.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Considere esta prova:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Consider this proof:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b1-b2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This is perfectly in keeping with the rules we have laid down already,
 and it should not seem particularly strange.
 Since `$B 
\backslash
eif B$' is a tautology, no particular premises should be required to prove
 it.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Isto está perfeitamente de acordo com as regras que já estabelecemos e,
 particularmente, não deveria parecer estranho.
 Uma vez que `
\begin_inset Formula $B\eif B$
\end_inset

' é uma tautologia, nenhuma premissa particular deveria ser exigida para
 prová-la.
 Mas suponha que tentássemos continuar a prova como se segue:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%But suppose we now tried to continue the proof as follows:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b1-b2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

tentativa imprópria
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

[
\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

x
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

de invocar 
\begin_inset Formula $\eif$
\end_inset

E
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con, b2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%If we were allowed to do this, it would be a disaster.
 It would allow us to prove any sentence letter from any other sentence
 letter.
 However, if you tell me that Anne is fast (symbolized by `$A$'), we shouldn't
 be able to conclude that Queen Boudica stood twenty-feet tall (symbolized
 by `$B$')! We must be prohibited from doing this, but how are we to implement
 the prohibition?
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Se fosse permitido fazer isto, seria um desastre.
 Seria permitido provar qualquer letra sentencial a partir de qualquer outra
 letra sentencial.
 Todavis, se você me diz que Anne é rápida (simbolizada por `
\begin_inset Formula $A$
\end_inset

'), não deveríamos ser capazes de concluir que a Rainha Boudica encontrava-se
 a seis metros de altura (simbolizada por `
\begin_inset Formula $B$
\end_inset

')! Temos de ser proibidos de fazer isto, mas como devemos implementar a
 proibição? 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We can describe the process of making an additional assumption as one of
 performing a 
\backslash
emph{subproof}: a subsidiary proof within the main proof.
 When we start a subproof, we draw another vertical line to indicate that
 we are no longer in the main proof.
  Then we write in the assumption upon which the subproof will be based.
 A subproof can be thought of as essentially posing this question: 
\backslash
emph{what could we show, if we also make this additional assumption?}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Podemos descrever o processo de fazer uma suposição adicional como o processo
 de executar uma 
\emph on
subprova
\emph default
: uma prova subsidiária [auxiliar] dentro da prova principal.
 Quando começamos uma subprova, desenhamos uma outra linha vertical para
 indicar que não estamos mais na prova principal.
 Então, escrevemos [na linha] da suposição na qual a subprova será baseada.
 Uma subprova pode ser pensada essencialmente como levantando esta questão:
 
\emph on
o que poderíamos mostrar, se fizermos também esta suposição adicional?
\emph default
 
\end_layout

\begin_layout Standard
Quando estamos trabalhando dentro da subprova, podemos referir-nos à suposição
 adicional que fizemos ao introduzir a subprova e a qualquer coisa que obtivemos
 a partir de nossas suposições iniciais (afinal das contas, estas suposições
 originais ainda estão valendo).
 Em algum momento, porém, desejaremos parar de trabalhar com a suposição
 adicional: desejaremos retornar da subprova para a prova principal.
 Para indicar que retornamos para a prova principal, a linha vertical da
 subprova termina.
 Neste ponto, dizemos que a subprova está 
\emph on
fechada
\emph default
.
 Depois de fechar uma subprova, deixamos de lado a suposição adicional e,
 assim, será ilegítimo inferir qualquer coisa que dependa desta suposição
 adicional.
 Assim, estipulamos: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 Para citar uma linha individual ao aplicar uma regra 
\end_layout

\begin_layout Enumerate
a linha deve vir antes da linha onde a regra é aplicada, mas
\end_layout

\begin_layout Enumerate
não ocorre dentro de uma subprova que foi fechada antes da linha onde a
 regra é aplicada
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Esta estipulação exclui a tentativa desastrosa de prova acima.
 A regra de 
\begin_inset Formula $\eif$
\end_inset

E requer que citemos duas linhas individuais anteriores na prova.
 Na suposta prova acima, uma das linhas (a saber, linha
\begin_inset space ~
\end_inset


\begin_inset Formula $4$
\end_inset

) ocorre dentro de uma subprova que foi fechada (na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $5$
\end_inset

).
 Isto é iligítimo.
 
\end_layout

\begin_layout Standard
O fechamento de uma suprova é chamado 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

descarte
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 das suposições desta subprova.
 Assim, podemos apresentar o ponto dessa forma: 
\emph on
voê não pode se referir a qualquer coisa que foi obtida usando suposições
 descartadas
\emph default
.
 
\end_layout

\begin_layout Standard
Subprovas permite-nos, então, a pensar sobre o que poderíamos mostrar, se
 fizéssemos suposições adicionais.
 A questão a ser tirada disto não é surpreendente — no curso de uma prova,
 temos de acompanhar cuidadosamente quais suposições estamos fazendo, em
 qualquer dado momento.
 Nosso sistema de prova faz isso de forma bastante gráfica (de fato, é precisame
nte por isso que escolhemos usar 
\emph on
este
\emph default
 sistema de prova).
 
\end_layout

\begin_layout Standard
Uma vez que começamos a pensar sobre o que podemos mostrar ao fazer suposições
 adicionais, nada nos impede de levantar a questão sobre o que poderíamos
 mostrar se fizéssemos 
\emph on
novas
\emph default
 suposições.
 Poderíamos ser motivados a introduzir uma subprova dentro de uma subprova.
 Aqui está um exemplo que apenas usa as regras de prova que consideramos
 até agora: 
\begin_inset ERT
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\backslash
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\backslash
hypo
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\end_inset


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{
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a
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}
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\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

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A
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
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{
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b
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}
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\end_inset


\begin_inset ERT
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{
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B
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}
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\backslash
open
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\backslash
hypo
\end_layout

\end_inset


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{
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c
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}
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{
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C
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\backslash
have
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ab
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}
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{
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A 
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 B
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\backslash
ai
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a,b
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{
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cab
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}
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{
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C 
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\backslash
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 (A 
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\backslash
eand
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 B)
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\backslash
ci
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c-ab
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bcab
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}
\end_layout

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\begin_inset ERT
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{
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B 
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\backslash
eif
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 (C 
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eif
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 (A 
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 B))
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\backslash
ci
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{
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b-cab
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}
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\begin_inset ERT
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\backslash
end{proof}
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\end_inset

 
\begin_inset ERT
status collapsed

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%Notice that the citation on line~$4$ refers back to the initial assumption
 (on line 1) and an assumption of a subproof (on line~$2$).
 This is perfectly in order, since neither assumption has been discharged
 at the time (i.e., by line~$4$).
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Perceba que a citação na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $4$
\end_inset

 refere-se à suposição inicial (na linha 1) e a uma suposição de uma subprova
 (na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $2$
\end_inset

).
 Isto está perfeitamente correto, uma vez que nenhuma suposição foi descartada
 ainda (o fechamento ocorre na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $4$
\end_inset

).
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Again, though, we need to keep careful track of what we are assuming at
 any given moment.
 Suppose we tried to continue the proof as follows:
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Novamente, porém, precisamos acompanhar cuidadosamente o que estamos assumindo
 em qualquer dado momento.
 Suponha que tentássemos continuar a prova como se segue: 
\begin_inset ERT
status collapsed

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\backslash
begin{proof}
\end_layout

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\backslash
hypo
\end_layout

\end_inset


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{
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a
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}
\end_layout

\end_inset


\begin_inset ERT
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{
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A
\begin_inset ERT
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}
\end_layout

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

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b
\begin_inset ERT
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}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

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B
\begin_inset ERT
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}
\end_layout

\end_inset

 
\begin_inset ERT
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\backslash
open
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\begin_inset ERT
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\backslash
hypo
\end_layout

\end_inset


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{
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c
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}
\end_layout

\end_inset


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{
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C
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\begin_inset ERT
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\backslash
have
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{
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ab
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}
\end_layout

\end_inset


\begin_inset ERT
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{
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A 
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\backslash
eand
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 B
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\backslash
ai
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{
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a,b
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\backslash
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have
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cab
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}
\end_layout

\end_inset


\begin_inset ERT
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{
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C 
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\backslash
eif
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\end_inset

 (A 
\begin_inset ERT
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\backslash
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 B)
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\end_inset


\begin_inset ERT
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\backslash
ci
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{
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c-ab
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\backslash
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\backslash
have
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{
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bcab
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}
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{
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B 
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\backslash
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\end_inset

(C 
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\backslash
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 (A 
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 B))
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\begin_inset ERT
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\backslash
ci
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\begin_inset ERT
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{
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b-cab
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\end_inset

 
\begin_inset ERT
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\backslash
have
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\begin_inset ERT
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{
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bcab
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}
\end_layout

\end_inset


\begin_inset ERT
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{
\end_layout

\end_inset

C 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B)
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
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\backslash
by
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\begin_inset ERT
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{
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tentativa imprópria
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}
\end_layout

\end_inset


\begin_inset ERT
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{}
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\begin_inset ERT
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\backslash
have
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\end_inset


\begin_inset ERT
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[
\end_layout

\end_inset

[
\begin_inset space \space{}
\end_inset


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]
\end_layout

\end_inset


\begin_inset ERT
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{
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x
\begin_inset ERT
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}
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

de invocar 
\begin_inset Formula $\eif$
\end_inset

I
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c-ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This would be awful.
 If we tell you that Anne is smart, you should not be able to infer that,
 if Cath is smart (symbolized by `$C$') then 
\backslash
emph{both} Anne is smart and Queen Boudica stood 20-feet tall! But this
 is just what such a proof would suggest, if it were permissible.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Isto seria horrível.
 Se lhe contarmos que Anne é esperta, você não deveria ser capaz de inferir
 que se Cath é esperta (simbolizada por `
\begin_inset Formula $C$
\end_inset

'), então Anne é esperta 
\emph on
e
\emph default
 a Rainha Boudica encontrava-se a 6 metros de altura.
 Mas isto é exatamente o que uma tal prova sugere, se isso fosse permitido.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The essential problem is that the subproof that began with the assumption~`$C$'
 depended crucially on the fact that we had assumed `$B$' on line~$2$.
 By line~$6$, we have 
\backslash
emph{discharged} the assumption~`$B$': we have stopped asking ourselves
 what we could show, if we also assumed `$B$'.
 So it is simply cheating, to try to help ourselves (on line~$7$) to the
 subproof that began with the assumption~`$C$'.
 Thus we stipulate, much as before, that a subproof can only be cited on
 a line if it does not occur within some other subproof which is already
 closed at that line.
  The attempted disastrous proof violates this stipulation.
 The subproof of lines $3$--$4$ occurs within a subproof that ends on line~$5$.
 So it cannot be invoked on line~$7$.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
O problema essencial é que a subprova que começava com a suposição
\begin_inset space ~
\end_inset

`
\begin_inset Formula $C$
\end_inset

' depende crucialmente do fato que tínhamos assumido `
\begin_inset Formula $B$
\end_inset

' na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $2$
\end_inset

.
 Na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $6$
\end_inset

, 
\emph on
descartamos
\emph default
 a suposição
\begin_inset space ~
\end_inset

`
\begin_inset Formula $B$
\end_inset

': paramos de nos perguntar o que poderíamos mostrar, se também assumíssemos
 `
\begin_inset Formula $B$
\end_inset

'.
 Assim, é simplesmente uma trapaça tentar obter a linha
\begin_inset space ~
\end_inset


\begin_inset Formula $7$
\end_inset

 da subprova que começa com a suposição
\begin_inset space ~
\end_inset

`
\begin_inset Formula $C$
\end_inset

'.
 Desse modo, estipulamos, como antes, que uma subprova pode apenas ser citada
 em uma linha se ela não ocorre dentro de alguma subprova que já está fechada
 nesta linha.
 A tentativa desastrosa de prova viola esta estipulação.
 A subprova das linhas 
\begin_inset Formula $3$
\end_inset

–
\begin_inset Formula $4$
\end_inset

 ocorre dentro de uma subprova que termina na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $5$
\end_inset

.
 Assim, ela não pode ser invocada na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $7$
\end_inset

.
\end_layout

\begin_layout Standard
Aqui está um outro caso que temos de excluir:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here is one further case we have to exclude:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
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\end_inset

b,c
\begin_inset ERT
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}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
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c2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bcab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

tentativa imprópria
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

[
\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

x
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

de invocar 
\begin_inset Formula $\eif$
\end_inset

I
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b-c2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here we are trying to cite a subproof that begins on line~$2$ and ends
 on line~$5$---but the sentence on line~$5$ depends not only on the assumption
 on line~$2$, but also on one another assumption (line~$3$) which we have
 not discharged at the end of the subproof.
 The subproof started on line~$3$ is still open at line~$3$.
 But $
\backslash
eif$I requires that the last line of the subproof 
\backslash
emph{only} relies on the assumption of the subproof being cited, i.e., the
 subproof beginning on line~$2$ (and anything before it), and not on assumptions
 of any subproofs within it.
 In particular, the last line of the subproof cited must not itself lie
 within a nested subproof.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Aqui estamos tentando citar uma subprova que começa na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $2$
\end_inset

 e termina na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $5$
\end_inset

 — mas a sentença na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $5$
\end_inset

 depende não só da suposição na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $2$
\end_inset

, mas também de uma outra suposição (linha
\begin_inset space ~
\end_inset


\begin_inset Formula $3$
\end_inset

) que não descartamos no fim da subprova.
 A subprova iniciada na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $3$
\end_inset

 ainda está aberta na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $3$
\end_inset

.
 Mas 
\begin_inset Formula $\eif$
\end_inset

I requer que a última linha da suprova conte 
\emph on
apenas
\emph default
 com a suposição da subprova a ser citada, ou seja, a subprova que inicia
 na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $2$
\end_inset

 (e qualquer coisa antes dela) e não conte com as suposições de quaisquer
 subprovas dentro dela.
 Em particular, a última linha da subprova citada não deve se encontrar
 dentro de uma subprova aninhada.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 Para citar uma subprova ao aplicar uma regra:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%To cite a subproof when applying a rule:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Enumerate
a subprova citada deve vir inteiramente antes da aplicação da regra onde
 ela é citada,
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%the cited subproof must come entirely before the application of the rule
 where it is cited, 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Enumerate
a subprova citada não deve se encontrar dentro de alguma outra subprova
 citada que está fechada na linha em que ela é citada, e
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%the cited subproof must not lie within some other closed subproof which
 is closed at the line it is cited, and 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Enumerate
a última linha da subprova citada não deve ocorrer dentro de uma subprova
 aninhada.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%last line of the cited subproof must not occur inside a nested subproof.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%One last point to emphasize how rules can be applied: where a rule requires
 you to cite an individual line, you cannot cite a subproof instead; and
 where it requires you to cite a subproof, you cannot cite an individual
 line instead.
 So for instance, this is incorrect:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Um último ponto que deve ser enfatizado é como as regras podem ser aplicadas:
 onde uma regra requer que você cite uma linha individual, você não pode
 citar uma subprova; e onde ela requer que você cite uma subprova, você
 não pode citar um linha individual.
 Desse modo, isto é incorreto: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b,c
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

tentativa imprópria
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

[
\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

x
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

de invocar R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c-c2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bcab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b-c3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here, we have tried to justify $C$ on line~$6$ by the reiteration rule,
 but we have cited the subproof on lines $3$--$5$ with it.
 That subproof is closed and can in principle be cited on line~$6$.
 (For instance, we could use it to justify $C 
\backslash
eif C$ by $
\backslash
eif$I.)  But the reiteration rule~R requires you to cite an individual line,
 so citing the entire subproof is inadmissible (even if that subproof contains
 the sentence~$C$ we want to reiterate).
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Aqui, tentamos justificar 
\begin_inset Formula $C$
\end_inset

 na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $6$
\end_inset

 pela regra de reiteração, mas citamos a subprova nas linhas
\begin_inset Formula $3$
\end_inset

–
\begin_inset Formula $5$
\end_inset

 com ela [sentença 
\begin_inset Formula $C$
\end_inset

].
 Esta subprova está fechada e, a princípio, pode ser citada na linha
\begin_inset space ~
\end_inset


\begin_inset Formula $6$
\end_inset

 (por exemplo, poderíamos usá-la para justificar 
\begin_inset Formula $C\eif C$
\end_inset

 por 
\begin_inset Formula $\eif$
\end_inset

I).
 Mas a regra de reiteração
\begin_inset space ~
\end_inset

R requer que você cite uma linha individual, assim citar a subprova inteira
 é inadmissível (mesmo se esta subprova contiver a sentença
\begin_inset space ~
\end_inset


\begin_inset Formula $C$
\end_inset

 que desejamos reiterar).
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%It is always permissible to open a subproof with any assumption.
 However, there is some strategy involved in picking a useful assumption.
 Starting a subproof with an arbitrary, wacky assumption would just waste
 lines of the proof.
 In order to obtain a conditional by {
\backslash
eif}I, for instance, you must assume the antecedent of the conditional in
 a subproof.
 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
É sempre permitido abrir uma subprova com qualquer suposição.
 Todavia, há alguma estrtégia envolvida na escolha de uma suposição útil.
 Começar uma subprova com uma suposição arbitrária e excêntrica apenas desperdiç
aria linhas da prova.
 A fim de obter, por exemplo, um condicional por meio de 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

I, devemos assumir o antecedente do condicional em uma subprova.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Equally, it is always permissible to close a subproof (and discharge its
 assumptions).
 However, it will not be helpful to do so until you have reached something
 useful.
 Once the subproof is closed, you can only cite the entire subproof in any
 justification.
 Those rules that call for a subproof or subproofs, in turn, require that
 the last line of the subproof is a sentence of some form or other.
  For instance, you are only allowed to cite a subproof for $
\backslash
eif$I if the line you are justifying is of the form $
\backslash
meta{A} 
\backslash
eif 
\backslash
meta{B}$, $
\backslash
meta{A}$ is the assumption of your subproof, and $
\backslash
meta{B}$ is the last line of your subproof.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Similarmente, é sempre permitido fechar uma subprova (e descartar as suposições
 dela).
 Entretanto, não será de qualquer ajuda fazer isto, até que você tenha chegado
 a algo útil.
 Uma vez que a subprova está fechada, você pode somente citar a subprova
 inteira em qualquer justificação.
 Aquelas regras que exigem uma subprova ou subprovas, por sua vez, requerem
 que a última linha da subprova seja uma sentença de alguma forma ou outra.
 Por exemplo, é somente permitido citar um subprova para 
\begin_inset Formula $\eif$
\end_inset

I, se a linha que você está justificando é da forma 
\begin_inset Formula $\meta{A}\eif\meta{B}$
\end_inset

, 
\begin_inset Formula $\meta{A}$
\end_inset

 é a suposição de sua subprova e 
\begin_inset Formula $\meta{B}$
\end_inset

 é a última linha de sua subprova.
\end_layout

\begin_layout Section
Biconditional
\end_layout

\begin_layout Standard
As regras para o bicondicional serão uma versão em duas etapas das regras
 para o condicional.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The rules for the biconditional will be like double-barrelled versions
 of the rules for the conditional.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%In order to prove `$W 
\backslash
eiff X$', for instance, you must be able to prove `$X$' on the assumption
 `$W$' 
\backslash
emph{and} prove `$W$' on the assumption `$X$'.
 The biconditional introduction rule ({
\backslash
eiff}I) therefore requires two subproofs.
 Schematically, the rule works like this:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A fim de provar `
\begin_inset Formula $W\eiff X$
\end_inset

', por exemplo, devemos ser capazes de provar `
\begin_inset Formula $X$
\end_inset

' supondo `
\begin_inset Formula $W$
\end_inset

' 
\emph on
e
\emph default
 de provar `
\begin_inset Formula $W$
\end_inset

' supondo `
\begin_inset Formula $X$
\end_inset

'.
 A regra de introdução do bicondicional (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eiff
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

I), portanto, requer duas subprovas.
 De forma esquemática, a regra trabalha dessa forma: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

i
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

j
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
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[
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l
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\begin_inset space \space{}
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ab
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meta
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\end_inset


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B
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bi
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{
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a1-b1,b2-a2
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\backslash
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}
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%There can be as many lines as you like between $i$ and $j$, and as many
 lines as you like between $k$ and $l$.
  Moreover, the subproofs can come in any order, and the second subproof
 does not need to come immediately after the first.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Podem existir tantas linhas quanto você quiser entre 
\begin_inset Formula $i$
\end_inset

 e 
\begin_inset Formula $j$
\end_inset

 e tantas linhas quanto você quiser entre 
\begin_inset Formula $k$
\end_inset

 e 
\begin_inset Formula $l$
\end_inset

.
 Ademais, as subprovas podem aparecer em qualquer ordem e a segunda subprova
 não precisa aparecer imediatamente depois da primeira.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The biconditional elimination rule ({
\backslash
eiff}E) lets you do a bit more than the conditional rule.
 If you have the left-hand subsentence of the biconditional, you can obtain
 the right-hand subsentence.
 If you have the right-hand subsentence, you can obtain the left-hand subsentenc
e.
 So we allow:
\end_layout

\end_inset


\end_layout

\begin_layout Standard
A regra de eliminação do bicondicional (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eiff
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

E) permite que você faça um pouco mais que a regra do condicional.
 Se você tem a subsentença à esquerda do bicondicional, você pode obter
 a subsentença à direita.
 Se você tem a subsentença à direita, você pode obter a subsentença à esquerda.
 Desse modo, é permitido: 
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\backslash
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{
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\backslash
begin{proof}
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have
\end_layout

\end_inset


\begin_inset ERT
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m
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]
\end_layout

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ab
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\begin_inset ERT
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ab,a
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b
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{
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A
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\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
be
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab,a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

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\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
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}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Note that the biconditional, and the right or left half, can be separated
 from one another, and they can appear in any order.
 However, in the citation for $
\backslash
eiff$E, we always cite the biconditional first.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Note que o bicondicional e a metade à direita ou a metade à esquerda podem
 ser separadas uma da outra e podem aparecer em qualquer ordem.
 Entretanto, na citação de 
\begin_inset Formula $\eiff$
\end_inset

E, sempre citamos o bicondicional primeiro.
\end_layout

\begin_layout Section
Disjunção
\end_layout

\begin_layout Standard
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status collapsed

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%Suppose Ludwig is reactionary.
 Then Ludwig is either reactionary or libertarian.
 After all, to say that Ludwig is either reactionary or libertarian is to
 say something weaker than to say that Ludwig is reactionary.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Suponha que Ludwig seja reacionário.
 Então, Ludwig é racionário ou libertário.
 Afinal da contas, dizer que Ludwig é reacionário ou libertário é dizer
 algo mais fraco do que dizer que Ludwig é reacionário.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Let's emphasize this point.
 Suppose Ludwig is reactionary.
 It follows that Ludwig is 
\backslash
emph{either} reactionary 
\backslash
emph{or} a kumquat.
 Equally, it follows that 
\backslash
emph{either} Ludwig is reactionary 
\backslash
emph{or} that kumquats are the only fruit.
  Equally, it follows that 
\backslash
emph{either} Ludwig is reactionary 
\backslash
emph{or} that God is dead.
  Many of these are strange inferences to draw, but there is nothing 
\backslash
emph{logically} wrong with them (even if they maybe violate all sorts of
 implicit conversational norms).
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Vamos enfatizar este ponto.Suponha que Ludwig seja reacionário.
 Segue-se que Ludwig é 
\emph on
ou
\emph default
 reacionário 
\emph on
ou
\emph default
 uma laranja.
 Da mesma maneira, segue-se que 
\emph on
ou
\emph default
 Ludwig é reacionário 
\emph on
ou
\emph default
 que laranjas são as únicas frutas.
 Similarmante, segue-se que 
\emph on
ou
\emph default
 Ludwig é reacionário 
\emph on
ou
\emph default
 que Deus está morto.
 Muitas destas [inferências] são inferências estranhas, mas há nada de 
\emph on
logicamente
\emph default
 errado com elas (mesmo se elas violem talvez todos os tipos de normas implícita
s de conversação).
 Armados com tudo isso, apresentamos a(s) regra(s) de introdução da disjunção:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Armed with all this, we present the disjunction introduction rule(s):
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


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status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

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\begin_inset ERT
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{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

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have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

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[
\end_layout

\end_inset

m
\begin_inset ERT
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\end_inset


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\end_inset


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\end_inset


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}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

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\end_layout

\end_inset


\begin_inset ERT
status collapsed

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}
\end_layout

\end_inset

 e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


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factoidbox
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
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\end_inset

 
\begin_inset ERT
status collapsed

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have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

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A
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\end_inset


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\end_inset


\begin_inset space \space{}
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\begin_inset ERT
status collapsed

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\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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ba
\begin_inset ERT
status collapsed

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}
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\end_inset


\begin_inset ERT
status collapsed

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meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

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\backslash
eor
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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A
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\end_inset


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}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Note que 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 pode ser 
\emph on
qualquer
\emph default
 sentença, assim o seguinte é perfeitamente aceitável:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Notice that 
\backslash
meta{B} can be 
\backslash
emph{any} sentence whatsoever, so the following is a perfectly acceptable
 proof:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

M
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

mmm
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

M 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 ([(A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eiff
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (C 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 D)] 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eiff
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

[E 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 F
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset

)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 Usar uma tabela de verdade para mostrar isto teria tomado 128 linhas.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Using a truth table to show this would have taken 128 lines.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The disjunction elimination rule is, though, slightly trickier.
 Suppose that either Ludwig is reactionary or he is libertarian.
 What can you conclude? Not that Ludwig is reactionary; it might be that
 he is libertarian instead.
 Equally, not that Ludwig is libertarian; for he might merely be reactionary.
 Disjunctions, just by themselves, are hard to work with.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Não obstante, a regra de eliminação da disjunção é uma pouco mais complicada.
 Suponha que Ludwig é reacionário ou ele é libertário.
 O que podemos concluir? Não podemos concluir que Ludwig seja reacionário;
 em vez disso, poderia ser o caso que ele seja libertário.
 Igualmente, não podemos concluir que Ludwig seja necessário; pois, ele
 poderia ser apenas reacionário.
 Disjunções em si são difíceis de se trabalhar.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%But suppose that we could somehow show both of the following: first, that
 Ludwig's being reactionary entails that he is an Austrian economist: second,
 that Ludwig's being libertarian entails that he is an Austrian economist.
 Then if we know that Ludwig is either reactionary or libertarian, then
 we know that, whichever he is, Ludwig is an Austrian economist.
 This insight can be expressed in the following rule, which is our disjunction
 elimination ($
\backslash
eor$E) rule:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Mas suponha que poderíamos, de alguma forma, mostrar ambas seguintes coisas:
 em primeiro lugar, o fato de Ludwig ser reacionário acarreta que ele seja
 um economista austríaco; em segundo lugar, o fato de Ludwig acarreta libertário
 que ele seja um economista austríaco; Então, se sabemos que Ludwig é ou
 reacionário ou libertário, então sabemos que, seja o que ele for, Ludwig
 é um economista austríaco.
 Este 
\emph on
insight
\emph default
 pode ser expresso na seguinte regra, que é a nossa regra de eliminação
 da disjunção (
\begin_inset Formula $\eor$
\end_inset

E): 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

i
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

j
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab, a-c1,b-c2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This is obviously a bit clunkier to write down than our previous rules,
 but the point is fairly simple.
 Suppose we have some disjunction, $
\backslash
meta{A} 
\backslash
eor 
\backslash
meta{B}$.
 Suppose we have two subproofs, showing us that $
\backslash
meta{C}$ follows from the assumption that $
\backslash
meta{A}$, and that $
\backslash
meta{C}$ follows from the assumption that $
\backslash
meta{B}$.
 Then we can infer $
\backslash
meta{C}$ itself.
 As usual, there can be as many lines as you like between $i$ and $j$, and
 as many lines as you like between $k$ and $l$.
 Moreover, the subproofs and the disjunction can come in any order, and
 do not have to be adjacent.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Isto é obviamente uma pouco mais abstruso de escrever do que nossas regras
 anteriores, mas o ponto é bastante simples.
 Suponha que tenhamos alguma disjunção 
\begin_inset Formula $\meta{A}\eor\meta{B}$
\end_inset

.
 Suponha que tenhamos duas subprovas, mostrando-nos que 
\begin_inset Formula $\meta{C}$
\end_inset

 se segue da suposição 
\begin_inset Formula $\meta{A}$
\end_inset

 e que 
\begin_inset Formula $\meta{C}$
\end_inset

 se segue da suposição 
\begin_inset Formula $\meta{B}$
\end_inset

.
 Então, podemos inferir 
\begin_inset Formula $\meta{C}$
\end_inset

.
 Como habitual, podem existir tantas linhas quanto você quiser entre 
\begin_inset Formula $i$
\end_inset

 e 
\begin_inset Formula $j$
\end_inset

 e podem existir tantas linhas quanto você quiser entre 
\begin_inset Formula $k$
\end_inset

 e 
\begin_inset Formula $l$
\end_inset

.
 Além disso, as subprovas e a disjunção podem aparecer em qualquer ordem
 e não precisam ser adjacentes.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%	
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Alguns exemplos poderiam ajudar a ilustrar isto:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Some examples might help illustrate this.
 Consider this argument:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset Formula 
\[
(P\eand Q)\eor(P\eand R)\therefore P
\]

\end_inset

Um exemplo de prova poderia ser executado assim:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%An example proof might run thus:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

prem
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 Q) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 (P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 R) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

pq
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 Q
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

p1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

pq
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

pr
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

p2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

pr
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

prem, pq-p1, pr-p2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 Aqui está um exemplo um pouco mais complexo.
 Considere:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here is a slightly harder example.
 Consider:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset Formula 
\[
A\eand(B\eor C)\therefore(A\eand B)\eor(A\eand C)
\]

\end_inset

Aqui está uma prova que corresponda a este argumento:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here is a proof corresponding to this argument:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

aboc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
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 (B 
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eor
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a
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A
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\backslash
ae
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aboc
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\backslash
have
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boc
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{
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B 
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\backslash
eor
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 C
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ae
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aboc
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\backslash
open
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\backslash
hypo
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B
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\backslash
have
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ab
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{
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A 
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\backslash
eand
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 B
\begin_inset ERT
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}
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\begin_inset ERT
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\backslash
ai
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\end_inset


\begin_inset ERT
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{
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a,b
\begin_inset ERT
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\end_inset

 
\begin_inset ERT
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\backslash
have
\end_layout

\end_inset


\begin_inset ERT
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{
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abo
\begin_inset ERT
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\begin_layout Plain Layout

}
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\begin_inset ERT
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{
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(A 
\begin_inset ERT
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\backslash
eand
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\end_inset

 B) 
\begin_inset ERT
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\backslash
eor
\end_layout

\end_inset

 (A 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C)
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
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{
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ab
\begin_inset ERT
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}
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\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
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\backslash
open
\end_layout

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\begin_inset ERT
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\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
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{
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c
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
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C
\begin_inset ERT
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\backslash
have
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ac
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}
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{
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A 
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\backslash
eand
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 C
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\backslash
ai
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{
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a,c
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\backslash
have
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aco
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\begin_inset ERT
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{
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(A 
\begin_inset ERT
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\backslash
eand
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\end_inset

 B) 
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\backslash
eor
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 (A 
\begin_inset ERT
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\backslash
eand
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 C)
\begin_inset ERT
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\backslash
oi
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close
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have
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(A 
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\backslash
eand
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 B) 
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\backslash
eor
\end_layout

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 (A 
\begin_inset ERT
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\backslash
eand
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 C)
\begin_inset ERT
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status open

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\backslash
oe
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status collapsed

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{
\end_layout

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boc, b-abo, c-aco
\begin_inset ERT
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}
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\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Don't be alarmed if you think that you wouldn't have been able to come
 up with this proof yourself.
 The ability to come up with novel proofs comes with practice, and we'll
 cover some strategies for finding proofs in 
\backslash
S
\backslash
ref{s:stratTFL}.
 The key question at this stage is whether, looking at the proof, you can
 see that it conforms to the rules that we have laid down.
 That just involves checking every line, and making sure that it is justified
 in accordance with the rules we have laid down.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Não fique assustado, se você acredita que não teria sido capaz de encontrar
 esta prova.
 A habilidade de encontrar novas provas vem com a prática e discutiremos
 algumas estratégias para achar provas em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:stratTFL"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 A questão chave neste estágio é se, olhando para a prova, podemos provar
 que ela está em conformidade com as regras que estabelecemos.
 Isto envolve apenas checar qualquer linha, certificando-se que ela está
 justificada de cordo com as regras que estabelecemos.
\end_layout

\begin_layout Section
Contradição e negação
\end_layout

\begin_layout Standard
Não tratamos apenas de um conectivo: a negação.
 Mas para lidar com esse conectivo, devemos conectar a negação com a 
\emph on
contradição
\emph default
.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We have only one connective left to deal with: negation.
 But to tackle it, we must connect negation with 
\backslash
emph{contradiction}.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%An effective form of argument is to argue your opponent into contradicting
 themselves.
 At that point, you have them on the ropes.
 They have to give up at least one of their assumptions.
 We are going to make use of this idea in our proof system, by adding a
 new symbol, `$
\backslash
ered$', to our proofs.
 This should be read as something like `contradiction!'
\backslash
 or `reductio!'
\backslash
 or `but that's absurd!' The rule for introducing this symbol is that we
 can use it whenever we explicitly contradict ourselves, i.e., whenever we
 find both a sentence and its negation appearing in our proof: 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Uma forma efetiva de argumento é mostrar que seu oponente caiu em contradição.
 Neste ponto, ele ficará nas cordas.
 Ele terá de desistir de pelo menos uma das suposições dele.
 Iremos usar esta ideia em nosso sistema de prova, adicionando um novo símbolo,
 `
\begin_inset Formula $\ered$
\end_inset

', às nossas provas.
 Isto deveria ser lido como `contradição!' ou `reductio!'
\begin_inset space \space{}
\end_inset

ou `mas isso é absurdo!'.
 A regra para introduzir este símolo é que podemos usá-lo, sempre que nos
 contradizemos explicitamente, ou seja, sempre que encontramos tanto uma
 sentença como a negação dela, que aparecem em nossa prova: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
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}
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
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[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

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]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
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[
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

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]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset

bot
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na, a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%It does not matter what order the sentence and its negation appear in,
 and they do not need to appear on adjacent lines.
 However, we always cite the line number of the negation first, followed
 by that of the sentence it is a negation of.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Não importa em qual ordem a sentença e a negação dela aparecem e elas não
 precisam aparecer em linhas adjacentes.
 Entretanto, sempre citamos o número da linha da negação primeiro, seguido
 pelo número da sentença da qual aquela é negação.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%There is obviously a tight link between contradiction and negation.
 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Há obviamente uma estreita ligação entre contradição e negação.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The rule $
\backslash
enot$E lets us proceed from two contradictory sentences---$
\backslash
meta{A}$ and its negation $
\backslash
enot 
\backslash
meta{A}$---to an explicit contradition~$
\backslash
ered$.
 We choose the label for a reason: it is the the most basic rule that lets
 us proceed from a premise containing a negation, i.e., $
\backslash
enot
\backslash
meta{A}$, to a sentence not containing it, i.e., $
\backslash
ered$.
 So it is a rule that 
\backslash
emph{eliminates}~$
\backslash
enot$.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A regra 
\begin_inset Formula $\enot$
\end_inset

E permite-nos prosseguir de duas sentenças contraditórias — 
\begin_inset Formula $\meta{A}$
\end_inset

 and its negation 
\begin_inset Formula $\enot\meta{A}$
\end_inset

 — para uma contradição explícita
\begin_inset space ~
\end_inset


\begin_inset Formula $\ered$
\end_inset

.
 Escolhemos o rótulo por uma razão: ela é a regra mais básica que nos permite
 prosseguir de uma premissa contendo uma negação, ou seja, 
\begin_inset Formula $\enot\meta{A}$
\end_inset

, para uma sentença que não a contém, ou seja, 
\begin_inset Formula $\ered$
\end_inset

.
 Desse modo, ela é uma regra que 
\emph on
elimina
\emph default

\begin_inset space ~
\end_inset


\begin_inset Formula $\enot$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We have said that `$
\backslash
ered$' should be read as something like `contradiction!' but this does not
 tell us much about the symbol.
 There are, roughly, three ways to approach the symbol.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Dissemos que `
\begin_inset Formula $\ered$
\end_inset

' deveria ser lido como `contradição!', mas isto não nos diz muito sobre
 o símbolo.
 Há, grosso modo, três maneiras de abordar o símbolo.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{ebullet}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Poderíamos considerar `
\begin_inset Formula $\ered$
\end_inset

' como uma nova sentença atômica de LVF, mas uma que pode ter apenas o valor
 de verdade Falso.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We might regard `$
\backslash
ered$' as a new atomic sentence of TFL, but one which can only ever have
 the truth value False.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Poderíamos considerar `
\begin_inset Formula $\ered$
\end_inset

' como uma abreviação para alguma contradição canônica tal como `
\begin_inset Formula $A\eand\enot A$
\end_inset

'.
 Isto terá o mesmo efeito como acima — obviamente, `
\begin_inset Formula $A\eand\enot A$
\end_inset

' tem apenas o valor de verdade Falso —, mas significa que, oficialmente,
 não precisamos adicionar um novo símbolo a LVF.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We might regard `$
\backslash
ered$' as an abbreviation for some canonical contradiction, such as `$A
 
\backslash
eand 
\backslash
enot A$'.
 This will have the same effect as the above---obviously, `$A 
\backslash
eand 
\backslash
enot A$' only ever has the truth value False---but it means that, officially,
 we do not need to add a new symbol to TFL.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset

Poderíamos considerar `
\begin_inset Formula $\ered$
\end_inset

' não como um símbolo de LVF, mas muito mais como um símbolo de pontuação
 que aparece na prova.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We might regard `$
\backslash
ered$', not as a symbol of TFL, but as something more like a 
\backslash
emph{punctuation mark} that appears in our proofs.
 (It is on a par with the line numbers and the vertical lines, say.)
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{ebullet}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%	
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%There is something very philosophically attractive about the third option,
 but here we will 
\backslash
emph{officially} adopt the first.
 `$
\backslash
ered$' is to be read as a sentence letter that is always false.
 This means that we can manipulate it, in our proofs, just like any other
 sentence.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Há algo filosoficamente atratvo sobre a terceira opção, mas aqui adotaremos
 
\emph on
oficialmente
\emph default
 a primeira.
 `
\begin_inset Formula $\ered$
\end_inset

' tem de ser lido como uma letra sentencial que é sempre falsa.
 Isto significa que podemos manipulá-la em nossas provas da mesma maneira
 que qualquer outra sentença.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We still have to state a rule for negation introduction.
 The rule is very simple: if assuming something leads you to a contradiction,
 then the assumption must be wrong.
 This thought motivates the following rule:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Ainda temos de enunciar uma regra para introdução da negação.
 A regra é muito simples: se assumir algo levá-o a uma contradição., então
 a suposição deve ser errada.
 Este pensamento motiva a seguinte regra: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

i
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

j
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a-nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%There can be as many lines between $i$ and $j$ as you like.
 To see this in practice, and interacting with negation, consider this proof:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Podem existir tantas linhas entre 
\begin_inset Formula $i$
\end_inset

 e 
\begin_inset Formula $j$
\end_inset

 quantas você quiser.
 Para ver isto na prática e interagir coma negação, considere esta prova:
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

d
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nd
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ndr
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nd, d
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nd-ndr
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% If the assumption that $
\backslash
meta{A}$ is true leads to a contradiction, $
\backslash
meta{A}$ cannot be true, i.e., it must be false, i.e., $
\backslash
enot
\backslash
meta{A}$ must be true.
 Of course, if the assumption that $
\backslash
meta{A}$ is false (i.e., the assumption that $
\backslash
enot
\backslash
meta{A}$ is true) leads to a contradiction, then $
\backslash
meta{A}$ cannot be false, i.e., $
\backslash
meta{A}$ must be true.
 So we can consider the following rule:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Se a suposição de que 
\begin_inset Formula $\meta{A}$
\end_inset

 é verdadeira leva a uma contradição, então 
\begin_inset Formula $\meta{A}$
\end_inset

 não pode ser verdadeira, ou seja, deve ser falsa, isto é, 
\begin_inset Formula $\enot\meta{A}$
\end_inset

 deve ser verdadeira.
 É claro, se a suposição de que 
\begin_inset Formula $\meta{A}$
\end_inset

 é falsa (ou seja, a suposição de que 
\begin_inset Formula $\enot\meta{A}$
\end_inset

 é verdadeira) leva a uma contradição, então 
\begin_inset Formula $\meta{A}$
\end_inset

 não pode ser falsa, isto é, 
\begin_inset Formula $\meta{A}$
\end_inset

 deve ser verdadeira.
 Desse modo, podemos considerar a seguinte regra: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

i
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

j
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ip
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a-nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This rule is called 
\backslash
emph{indirect proof}, since it allows us to prove $
\backslash
meta{A}$ indirectly, by assuming its negation.
 Formally, the rule is very similar to $
\backslash
enot$I, but $
\backslash
meta{A}$ and $
\backslash
enot
\backslash
meta{A}$ have changed places.
 Since $
\backslash
enot
\backslash
meta{A}$ is not the conclusion of the rule, we are not introducing~$
\backslash
enot$, so IP is not a rule that introduces any connective.
 It also doesn't eliminate a connective, since it has no free-standing premises
 which contain~$
\backslash
enot$, only a subproof with an assumption of the form~$
\backslash
enot
\backslash
meta{A}$.
  By contrast, $
\backslash
enot$E does have a premise of the form $
\backslash
enot
\backslash
meta{A}$: that's why $
\backslash
enot$E eliminates~$
\backslash
enot$, but IP does not
\backslash
footnote{There are logicians who have qualms about IP, but not about $
\backslash
enot$E.
 They are called ``intuitionists.'' Intuitionists don't buy our basic assumption
 that every sentence has one of two truth values, true or false.
 They also think that $
\backslash
enot$ works differently---for them, a proof of $
\backslash
ered$ from $
\backslash
meta{A}$ guarantees $
\backslash
enot 
\backslash
meta{A}$, but a proof of $
\backslash
ered$ from $
\backslash
enot
\backslash
meta{A}$ does not guarantee that~$
\backslash
meta{A}$, but only $
\backslash
enot
\backslash
enot
\backslash
meta{A}$.
 So, for them, $
\backslash
meta{A}$ and $
\backslash
enot
\backslash
enot
\backslash
meta{A}$ are not equivalent.}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Esta regra é chama 
\emph on
prova indireta
\emph default
, uma vez que ela permite que provemos 
\begin_inset Formula $\meta{A}$
\end_inset

 indiretamente, assumindo a negação dessa sentença.
 Formalmente, a regra é muito similar a 
\begin_inset Formula $\enot$
\end_inset

I, mas 
\begin_inset Formula $\meta{A}$
\end_inset

 e 
\begin_inset Formula $\enot\meta{A}$
\end_inset

 mudam de posições.
 Uma vez que 
\begin_inset Formula $\enot\meta{A}$
\end_inset

 não é a conclusão da regra, não estamos introduzindo 
\begin_inset space ~
\end_inset


\begin_inset Formula $\enot$
\end_inset

, assim IP não é uma regra que introduz qualquer conectivo.
 Ela não elimina também um conectivo, uma vez que ela não tem premissas
 independentes que contenha
\begin_inset space ~
\end_inset


\begin_inset Formula $\enot$
\end_inset

, tem apenas uma subprova com uma suposição da forma
\begin_inset space ~
\end_inset


\begin_inset Formula $\enot\meta{A}$
\end_inset

.
 Por outro lado, 
\begin_inset Formula $\enot$
\end_inset

E tem uma premissa da forma 
\begin_inset Formula $\enot\meta{A}$
\end_inset

: é por isso que 
\begin_inset Formula $\enot$
\end_inset

E elimina
\begin_inset space ~
\end_inset


\begin_inset Formula $\enot$
\end_inset

, mas IP não elimina
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
|Há lógicos que têm reservas em relação a IP, mas não em relação a 
\begin_inset Formula $\enot$
\end_inset

E.
 Eles são chamados 
\begin_inset Quotes fls
\end_inset

intuicionistas
\begin_inset Quotes frs
\end_inset

.
 Intuicionistas não compram nossas suposições básicas que toda sentença
 tem um dos dois valores de verdade, verdadeiro ou falso.
 Eles também pensam que 
\begin_inset Formula $\enot$
\end_inset

 funciona de forma diferente — para eles, uma prova de 
\begin_inset Formula $\ered$
\end_inset

 a partir de 
\begin_inset Formula $\meta{A}$
\end_inset

 garante 
\begin_inset Formula $\enot\meta{A}$
\end_inset

, mas uma prova de 
\begin_inset Formula $\ered$
\end_inset

 a partir de 
\begin_inset Formula $\enot\meta{A}$
\end_inset

 não garante que
\begin_inset space ~
\end_inset


\begin_inset Formula $\meta{A}$
\end_inset

, mas somente 
\begin_inset Formula $\enot\enot\meta{A}$
\end_inset

.
 Portanto, para eles, 
\begin_inset Formula $\meta{A}$
\end_inset

 e 
\begin_inset Formula $\enot\enot\meta{A}$
\end_inset

 não são equivalentes.
\end_layout

\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Using $
\backslash
enot$I, we were able to give a proof of $
\backslash
enot
\backslash
enot
\backslash
meta{D}$ from $
\backslash
meta{D}$.
 Using IP, we can go the other direction (with essentially the same proof).
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Usando 
\begin_inset Formula $\enot$
\end_inset

I, fomos capazes de dar uma prova de of 
\begin_inset Formula $\enot\enot\meta{D}$
\end_inset

 a partir de 
\begin_inset Formula $\meta{D}$
\end_inset

.
 Usando IP, podemos ir na direção contrária (essencialmente a prova é a
 mesma).
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

d
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nd
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ndr
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

d, nd
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ip
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nd-ndr
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We need one last rule.
 It is a kind of elimination rule for `$
\backslash
ered$', and known as 
\backslash
emph{explosion}.
 
\backslash
footnote{The latin name for this principle is 
\backslash
emph{ex contradictione quod libet}, ``from contradiction, anything.''} If
 we obtain a contradiction, symbolized by `$
\backslash
ered$', then we can infer whatever we like.
 How can this be motivated, as a rule of argumentation? Well, consider the
 English rhetorical device `
\backslash
ldots and if 
\backslash
emph{that's} true, I'll eat my hat'.
 Since contradictions simply cannot be true, if one 
\backslash
emph{is} true then not only will I eat my hat, I'll have it too.
 Here is the formal rule:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Precisamos de uma última regra.
 É um tipo de regra de eliminação para `
\begin_inset Formula $\ered$
\end_inset

' e é conhecida como 
\emph on
explosão
\emph default

\begin_inset Foot
status collapsed

\begin_layout Plain Layout
O nome latin para este princípio é 
\emph on
ex contradictione quod libet
\emph default
, 
\begin_inset Quotes fls
\end_inset

de uma contradição, [segue-se] qualquer coisa
\begin_inset Quotes frs
\end_inset

.
\end_layout

\end_inset

.
 Se obtemos uma contradição, simbolizada por `
\begin_inset Formula $\ered$
\end_inset

', então podemos inferir qualquer coisa que quisermos.
 Como isto pode ser motivado, como uma regra de argumentação? Ora, considere
 o estratagema retórico do Português `…e se 
\emph on
isso
\emph default
 é verdadeiro, eu comerei meu chapéu'.
 Uma vez que, simplesmente, contradições não podem ser verdadeiras, se alguma
 [contradição] 
\emph on
for
\emph default
 verdadeira, então não só comerei meu chapéu como também ficarei com ele.
 Aqui está a regra formal: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
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\end_inset


\begin_inset ERT
status collapsed

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{
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\begin_inset ERT
status collapsed

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\backslash
begin{proof}
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\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
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\end_inset


\begin_inset ERT
status collapsed

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[
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\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bot
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
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\end_inset


\begin_inset ERT
status collapsed

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}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
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\end_inset


\begin_inset ERT
status collapsed

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[
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\end_inset

 
\begin_inset ERT
status collapsed

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]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset


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status collapsed

\begin_layout Plain Layout


\backslash
meta
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
re
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bot
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Note que 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 pode ser 
\emph on
qualquer
\emph default
 sentença.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Note that 
\backslash
meta{A} can be 
\backslash
emph{any} sentence whatsoever.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The explosion rule is a bit odd.
 It looks like 
\backslash
meta{A} arrives in our proof like a bunny out of a hat.
 When trying to find proofs, it is very tempting to try to use it everywhere,
 since it seems so powerful.
  Resist this temptation: you can only apply it when you already have~$
\backslash
ered$!  And you get $
\backslash
ered$ only when your assumptions are contradictory.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A regra de explosão é um pouco estranha.
 Parece que 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 ocorre na prova como que do nada.
 Ao tentar encontrar provas, é muito tentador usá-la em qualquer lugar,
 uma vez que ela parece ser muito poderosa.
 Resista a esta tentação: você apenas pode aplicá-la quando você já tiver
\begin_inset space ~
\end_inset


\begin_inset Formula $\ered$
\end_inset

! E você obtém 
\begin_inset Formula $\ered$
\end_inset

 somente quando suas suposições são contraditórias.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Still, isn't it odd that from a contradiction anything whatsoever should
 follow? Not according to our notion of entailment and validity.
 For 
\backslash
meta{A} entails 
\backslash
meta{B} iff there is no valuation of the sentence letters which makes 
\backslash
meta{A} true and 
\backslash
meta{B} false at the same time.
 Now $
\backslash
ered$ is a contradiction---it is never true, whatever the valuation of the
 sentence letters.
 Since there is no valuation which makes $
\backslash
ered$ true, there of course is also no valuation that makes $
\backslash
ered$ true and 
\backslash
meta{B} false! So according to our definition of entailment, $
\backslash
ered 
\backslash
entails 
\backslash
meta{B}$, whatever 
\backslash
meta{B} is.
 A contradiction entails anything.
 
\backslash
footnote{There are some logicians who don't buy this.
 They think that if 
\backslash
meta{A} entails 
\backslash
meta{B}, there must be some 
\backslash
emph{relevant connection} between 
\backslash
meta{A} and 
\backslash
meta{B}---and there isn't one between $
\backslash
ered$ and some arbitrary sentence~
\backslash
meta{B}.
 So these logicians develop other, ``relevant'' logics in which you aren't
 allowed the explosion rule.}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Ainda assim, não é estranho que de uma contradição deveria seguir-se qualquer
 coisa? Não de acordo com a nossa noção de acarretamento e validade.
 Pois, 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 acarreta 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 sse não há valoração da letras sentenciais que faça 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 verdadeira e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 falsa ao mesmo tempo.
 Ora, 
\begin_inset Formula $\ered$
\end_inset

 é uma contradição — nunca é verdadeira, qualquer que seja a valoração das
 letras sentenciais.
 Uma vez que não há valoração que faça 
\begin_inset Formula $\ered$
\end_inset

 verdadeira, não há também, é claro, valoração que faça 
\begin_inset Formula $\ered$
\end_inset

 verdadeira e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 falsa! Assim, de acordo com nossa definição de acarretamento, 
\begin_inset Formula $\ered\entails\meta{B}$
\end_inset

, seja o que for 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 Uma contradição acarreta tudo
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Há alguns lógicos que não compram isto.
 Eles pensam que se 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 acarreta 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

, deve existir alguma 
\emph on
conexão relevante
\emph default
 entre 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 — e não há entre 
\begin_inset Formula $\ered$
\end_inset

 e algumas sentença arbitrária
\begin_inset space ~
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 Desse modo, estes lógicos devenvolvem outras lógicas, chamadas 
\begin_inset Quotes fls
\end_inset

relevantes
\begin_inset Quotes frs
\end_inset

, nas quais a regra de explosão não é permitida.
\end_layout

\end_inset

.
\end_layout

\begin_layout Standard

\emph on
Estas são todas as regras básicas para o sistema de prova de LVF.

\emph default
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
emph{These are all of the basic rules for the proof system for TFL.}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
practiceproblems
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\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 As seguintes duas `provas' estão 
\emph on
incorretas
\emph default
.
 Explique os erros cometidos.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The following two `proofs' are 
\backslash
emph{incorrect}.
 Explain the mistakes they make.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

abc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 L 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 A) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

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nla
\begin_inset ERT
status collapsed

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}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 L 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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nl
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nl
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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\end_inset

abc
\begin_inset ERT
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}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
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\begin_inset ERT
status collapsed

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\backslash
open
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\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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red
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status collapsed

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}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nl, l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
re
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
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red
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

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A
\begin_inset ERT
status collapsed

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}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
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\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

abc, nla-a, l-a2
\begin_inset ERT
status collapsed

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}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
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status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
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\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

abc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

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bcd
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status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

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(B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C) 
\begin_inset ERT
status collapsed

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\backslash
eif
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\end_inset

 D
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}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

abc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

d
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ce{bc, bcd}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Estão faltando citações (regra e números de linha) nas seguintes três provas.
 Adicione-as para transformá-las em provas 
\emph on
bona fide
\emph default
.
 Além disso, escreva o argumento que corresponda a cada prova.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The following three proofs are missing their citations (rule and line numbers).
 Add them, to turn them into 
\backslash
emph{bona fide} proofs.
 Additionally, write down the argument that corresponds to each proof.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{multicols}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ps
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 S
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nsor
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

S 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

p
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
ae{ps}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

s
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

S
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
ae{ps}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

r
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
ce{nsor, s}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

re
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 E
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
oi{r}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
par 
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ad
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
ae{ab}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

d
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
ce{ad, a}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

de
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

D 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 E
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
oi{d}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

conc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (D 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 E)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
ci{ab-de}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
par 
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nlcjol
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 L 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (J 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 L)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nl
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

jol
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

J 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
ce{nlcjol, nl}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

j
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

J
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

jj
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

J 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 J
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
ai{j}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

j2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

J
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
ae{jj}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

L
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

red
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
ne{nl, l}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

j3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

J
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
re{red}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

conc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

J
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
oe{jol, j-j2, l-j3}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{multicols}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
solutions
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "pr.solvedTFLproofs"

\end_inset

 Dê uma prova para cada um dos seguintes argumentos:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Give a proof for each of the following arguments:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $J\eif\enot J\therefore\enot J$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $Q\eif(Q\eand\enot Q)\therefore\enot Q$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif(B\eif C)\therefore(A\eand B)\eif C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $K\eand L\therefore K\eiff L$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(C\eand D)\eor E\therefore E\eor D$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eiff B,B\eiff C\therefore A\eiff C$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot F\eif G,F\eif H\therefore G\eor H$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(Z\eand K)\eor(K\eand M),K\eif D\therefore D$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $P\eand(Q\eor R),P\eif\enot R\therefore Q\eor E$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $S\eiff T\therefore S\eiff(T\eor S)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(P\eif Q)\therefore\enot Q$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(P\eif Q)\therefore P$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Chapter
Construindo provas
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "s:stratTFL"

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Constructing proofs
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%There is no simple recipe for finding proofs, and there is no substitute
 for practice.
 Here, though, are some rules of thumb and strategies to keep in mind.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Não existe receita simples para encontrar provas e não há substituto para
 a prática.
 Aqui, todavia, estão algumas regras de ouro e estratégias para sempre ter
 em mente.
\end_layout

\begin_layout Section
Trabalhando de trás para frente
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working backward from what we want
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%So you're trying to find a proof of some conclusion~$
\backslash
meta{C}$, which will be the last line of your proof.
 The first thing you do is look at~$
\backslash
meta{C}$ and ask what the introduction rule is for its main logical operator.
 This gives you an idea of what should happen 
\backslash
emph{before} the last line of the proof.
 The justifications for the introduction rule require one or two other sentences
 above the last line, or one or two subproofs.
 Moreover, you can tell from~$
\backslash
meta{C}$ what those sentences are, or what the assumptions and conclusions
 of the subproof(s) are.
 Then you can write down those sentence or outline the subproof(s) above
 the last line, and treat those as your new goals.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Suponha que você esteja tentando encontrar uma prova de alguma conclusão
\begin_inset space ~
\end_inset


\begin_inset Formula $\meta{C}$
\end_inset

, que será a última linha da sua prova.
 A primeira coisa a fazer é olhar para 
\begin_inset Formula $\meta{C}$
\end_inset

 e perguntar qual é a regra de introdução para o operador lógico principal
 dela.
 Isto já dá uma ideia do que deveria acontecer 
\emph on
antes
\emph default
 da última linha da prova.
 As justificações para a regra de introdução exige uma ou duas sentenças
 acima da última linha, ou uma ou duas subprovas.
 Além disso, a partir de 
\begin_inset Formula $\meta{C}$
\end_inset

, você pode falar quais são aquelas sentenças ou quais são as suposições
 e conclusões da(s) subprova(s).
 Então, podemos escrever aquelas sentenças ou esboçar as subprova(s) acima
 da última linha e tratar tudo isso como sendo seus novos objetivos.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%For example: If your conclusion is a conditional $
\backslash
meta{A}
\backslash
eif
\backslash
meta{B}$, plan to use the {
\backslash
eif}I rule.
 This requires starting a subproof in which you assume~
\backslash
meta{A}.
 The subproof ought to end with~
\backslash
meta{B}.
 Then, continue by thinking about what you should do to get $
\backslash
meta{B}$ inside that subproof, and how you can use the assumption~$
\backslash
meta{A}$.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Por exemplo: se sua conclusão é um condicional 
\begin_inset Formula $\meta{A}\eif\meta{B}$
\end_inset

, planeje usar a regra 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

I.
 Isto requer que você comece uma subprova na qual você assuma 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

..
 A subprova deveria terminar com 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 Então, continue pensando sobre o que você deveria fazer para obter 
\begin_inset Formula $\meta{B}$
\end_inset

 dentro desta subprova e como você pode usar a suposição 
\begin_inset Formula $\meta{A}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%If your goal is a conjunction, conditional, or negated sentence, you should
 start by working backward in this way.
 We'll describe what you have to do in each of these cases in detail.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Se seu objetivo é uma conjunção, um condicional ou uma sentença negada,
 você deveria começar trabalhando de trás para frente.
 Descreveremos em detalhes o que você tem de fazer em cada um de stes casos.
\end_layout

\begin_layout Subsection*
Trabalhando de trás para frente a partir de uma conjunção
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working backward from a conjunction
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%If we want to prove $
\backslash
meta{A} 
\backslash
eand 
\backslash
meta{B}$, working backward means we should write $
\backslash
meta{A} 
\backslash
eand 
\backslash
meta{B}$ at the bottom of our proof, and try to prove it using $
\backslash
eand$I.
 At the top, we'll write out the premises of the proof, if there are any.
 Then, at the bottom, we write the sentence we want to prove.
 If it is a conjunction, we'll prove it using $
\backslash
eand$I.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Se quisermos provar 
\begin_inset Formula $\meta{A}\eand\meta{B}$
\end_inset

, trabalhar de trás para frente significa que deveríamos escrever 
\begin_inset Formula $\meta{A}\eand\meta{B}$
\end_inset

 na parte inferior da nossa prova e tentar prová-la usando 
\begin_inset Formula $\eand$
\end_inset

I.
 No topo, escreveremos as premissas da prova, se existirem.
 Então, na parte inferior, escrevemos a sentença que queremos provar.
 Se ela for uma conjunção, prová-la-emos usando 
\begin_inset Formula $\eand$
\end_inset

I.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

n,m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%For $
\backslash
eand$I, we need to prove $
\backslash
meta{A}$ first, then prove $
\backslash
meta{B}$.
 For the last line, we have to cite the lines where we (will have) proved
 $
\backslash
meta{A}$ and  $
\backslash
meta{B}$, and use~$
\backslash
eand$I.
 The parts of the proof labelled $
\backslash
vdots$ have to still be filled in.
 We'll mark the line numbers $m$, $n$ for now.
 When the proof is complete, these placeholders can be replaced by actual
 numbers.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Para 
\begin_inset Formula $\eand$
\end_inset

I, precisamos provar 
\begin_inset Formula $\meta{A}$
\end_inset

 primeiro, então provar 
\begin_inset Formula $\meta{B}$
\end_inset

.
 Na última linha, temos de citar as linhas onde provamos (teremos provado)
 
\begin_inset Formula $\meta{A}$
\end_inset

 e 
\begin_inset Formula $\meta{B}$
\end_inset

, e usamos 
\begin_inset Formula $\eand$
\end_inset

I
\begin_inset Formula $m$
\end_inset

, 
\begin_inset Formula $n$
\end_inset

.
 As partes da prova rotuladas por 
\begin_inset Formula $\vdots$
\end_inset

 ainda têm de ser preenchidas.
 Marcaremos os números de linha 
\begin_inset Formula $m$
\end_inset

, 
\begin_inset Formula $n$
\end_inset

 por enquanto.
 Quando a prova estiver completa, estas metavariáveis podem ser substituídas
 por números reais.
\end_layout

\begin_layout Subsection*
Trabalhando de trás para frente a partir de um condicional
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working backward from a conditional
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%If our goal is to prove a conditional $
\backslash
meta{A} 
\backslash
eif 
\backslash
meta{B}$, we'll have to use $
\backslash
eif$I.
 This requires a subproof starting with $
\backslash
meta{A}$ and ending with~$
\backslash
meta{B}$.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Se nosso objetivo é provar um condicional 
\begin_inset Formula $\meta{A}\eif\meta{B}$
\end_inset

, teremos de usar 
\begin_inset Formula $\eif$
\end_inset

I.
 Isto requer uma subprova iniciando com 
\begin_inset Formula $\meta{A}$
\end_inset

 e terminando com 
\begin_inset Formula $\meta{B}$
\end_inset

.
\end_layout

\begin_layout Standard
Estabeleceremos nossa prova da seguinte forma:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We'll set up our proof as follows:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 Novamente deixaremos metavariáveis `
\begin_inset Formula $m$
\end_inset

' e `
\begin_inset Formula $n$
\end_inset

' como reservadas para o número de linha.
 Registraremos a última inferência como 
\begin_inset Formula $\eif$
\end_inset

I, citando a subprova.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Again we'll leave placeholders in the line number slots.
 We'll record the last inference as $
\backslash
eif$I, citing the subproof.
\end_layout

\end_inset


\end_layout

\begin_layout Subsection*
Trabalhando de trás para frente a partir de uma sentença negada
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working backward from a negated sentence
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Se quisermos provar 
\begin_inset Formula $\enot\meta{A}$
\end_inset

, teremos de usar 
\begin_inset Formula $\enot$
\end_inset

I.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% If we want to prove $
\backslash
enot 
\backslash
meta{A}$, we'll have to use $
\backslash
enot$I.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%For $
\backslash
enot$I, we have to start a subproof with assumption $
\backslash
meta{A}$; the last line of the subproof has to be $
\backslash
ered$.
  We'll cite the subproof, and use~$
\backslash
enot$I as the rule.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Para 
\begin_inset Formula $\enot$
\end_inset

I, temos de iniciar uma subprova com suposição 
\begin_inset Formula $\meta{A}$
\end_inset

; a última linha da subprova tem de ser 
\begin_inset Formula $\ered$
\end_inset

.
 Citaremos a subprova e usaremos 
\begin_inset Formula $\enot$
\end_inset

I como a regra.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%When working backward, continue to do so as long as you can.
 So if you're working backward to prove $
\backslash
meta{A} 
\backslash
eif 
\backslash
meta{B}$ and have set up a subproof in which you want to prove $
\backslash
meta{B}$.
 Now look at~$
\backslash
meta{B}$.
 If, say, it is a conjunction, work backward from it, and write down the
 two conjuncts inside your subproof.
 Etc.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Ao trabalhar de trás para frente, continue fazendo isso até onde puder.
 Desse modo, se você está trabalhando de trás para frente para provar 
\begin_inset Formula $\meta{A}\eif\meta{B}$
\end_inset

, estabeleça uma subprova na qual você você deseja provar 
\begin_inset Formula $\meta{B}$
\end_inset

.
 Olhe agora para 
\begin_inset Formula $\meta{B}$
\end_inset

.
 Se, digamos, ela for uma conjunção, trabalhe de trás para frente a partir
 dela e escreva os dois conjuntos dentro da subprova etc.
\end_layout

\begin_layout Subsection*
Trabalhando de trás para frente a partir de uma disjunção
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working backward from a disjunction
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Of course, you can also work backward from a disjunction $
\backslash
meta{A} 
\backslash
eor 
\backslash
meta{B}$, if that is your goal.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

É claro, você também pode trabalhar de trás para frente a partir de uma
 disjunção 
\begin_inset Formula $\meta{A}\eor\meta{B}$
\end_inset

, se isso é seu objetivo.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The $
\backslash
eor$I rule requires that you have one of the disjuncts in order to infer
 $
\backslash
meta{A} 
\backslash
eor 
\backslash
meta{B}$.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A regra 
\begin_inset Formula $\eor$
\end_inset

I requer que você tenha um dos disjuntos a fim de inferir 
\begin_inset Formula $\meta{A}\eor\meta{B}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%So to work backward, you pick a disjunct, infer $
\backslash
meta{A} 
\backslash
eor 
\backslash
meta{B}$ from it, and then continue to look for a proof of the disjunct
 you picked:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Desse modo, para trabalhar de trás para frente, escolha um disjunto e infira
 
\begin_inset Formula $\meta{A}\eor\meta{B}$
\end_inset

 dele e, então, continue buscando uma prova do disjunto que você escolheu:
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%However, you may not be able to prove the disjunct you picked.
 In that case you have to backtrack.
 When you can't fill in the $
\backslash
vdots$, delete everything, and try with the other disjunct:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Todavia, pode acontecer de você não ser capaz de provar o disjunto que você
 escolher.
 Neste caso, você tem de dar um passo atrás.
 Quando você não consegue preencher o 
\begin_inset Formula $\vdots$
\end_inset

, exclua tudo e tente com o outro disjunto: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Obviously, deleting everything and starting over is frustrating, so you
 should avoid it.
 If your goal is a disjunction, therefore, you should 
\backslash
emph{not start} by working backward: try working forward first, and apply
 the $
\backslash
eor$I strategy only when working forward (and working backward using $
\backslash
eand$I, $
\backslash
eif$I, and $
\backslash
enot$I) no longer work.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Obviamente, excluir tudo e recomeçar é frustrante, assim você deveria evitar
 isso.
 Se, portanto, seu objetivo é uma disjunção, você 
\emph on
não
\emph default
 deveria 
\emph on
iniciar
\emph default
 trabalhando de trás para frente: tente trabalhar de forma direta primeiro
 e aplique a estratégia 
\begin_inset Formula $\eor$
\end_inset

I apenas quando trabalhar diretamente (e trabalhar de trás para frente usando
 
\begin_inset Formula $\eand$
\end_inset

I, 
\begin_inset Formula $\eif$
\end_inset

I e 
\begin_inset Formula $\enot$
\end_inset

I) não funcionam mais.
\end_layout

\begin_layout Section
Trabalhe diretamente a partir do que você tem
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Work forward from what you have
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Your proof may have premises.
 And if you've worked backward in order to prove a conditional or a negated
 sentence, you will have set up subproofs with an assumption, and be looking
 to prove a final sentence in the subproof.
 These premises and assumptions are sentences you can work forward from
 in order to fill in the missing steps in your proof.
 That means applying elimination rules for the main operators of these sentences.
 The form of the rules will tell you what you'll have to do.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Sua prova pode ter premissas.
 E se você trabalhar de trás para frente a fim de provar um condicional
 ou uma sentença negada, você estabelecerá subprovas com uma suposição e
 buscará provar um sentença final na subprova.
 Estas premissas e suposições são sentenças que podemos trabalhar diretamente
 a fim de preencher os passos perdidos em sua prova.
 Isso significa aplicar regras de eliminação para os operadores principais
 destas sentenças.
 A forma das regras dir-lhe-ão o que você terá de fazer.
\end_layout

\begin_layout Subsection*
Trabalhando diretamente a partir de uma conjunção
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working forward from a conjunction
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%To work forward from a sentence of the form $
\backslash
meta{A} 
\backslash
eand 
\backslash
meta{B}$, we use $
\backslash
eand$E.
 That rule allows us to do two things: infer $
\backslash
meta{A}$, and infer $
\backslash
meta{B}$.
 So in a proof where we have $
\backslash
meta{A} 
\backslash
eand 
\backslash
meta{B}$, we can work forward by writing $
\backslash
meta{A}$ and/or $
\backslash
meta{B}$ immediately below the conjunction: 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Para trabalhar diretamente a partir de uma sentença da forma 
\begin_inset Formula $\meta{A}\eand\meta{B}$
\end_inset

, usamos 
\begin_inset Formula $\eand$
\end_inset

E.
 Esta regra permite que façamos duas coisas: inferir 
\begin_inset Formula $\meta{A}$
\end_inset

 e inferir 
\begin_inset Formula $\meta{B}$
\end_inset

.
 Assim, em uma prova onde temos 
\begin_inset Formula $\meta{A}\eand\meta{B}$
\end_inset

, podemos trabalhar diretamente, inferindo 
\begin_inset Formula $\meta{A}$
\end_inset

 or 
\begin_inset Formula $\meta{B}$
\end_inset

 (ou ambos) imediatamente abaixo da conjunção.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Usually it will be clear in the particular situation you're in which one
 of 
\backslash
meta{A} or 
\backslash
meta{B} you'll need.
 It doesn't hurt, however, to write them both down.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Em geral, ficará claro , por conta da situação particular, de qual dos dois
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 or 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 você precisará.
 Mas não há qualquer dano escrever ambos.
\end_layout

\begin_layout Subsection*
Trabalhando diretamente a partir de uma disjunção
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working forward from a disjunction
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working forward from a disjunction works a bit differently.
 To use a disjunction, we use the $
\backslash
eor$E rule.
 In order to apply that rule, it is not enough to know what the disjuncts
 of the disjunction are that we want to use.
 We must also keep in mind what we want to prove.
 Suppose we want to prove~$
\backslash
meta{C}$, and we have $
\backslash
meta{A} 
\backslash
eor B$ to work with.
 (That $
\backslash
meta{A} 
\backslash
eor B$ may be a premise of the proof, an assumption of a subproof, or something
 already proved.) In order to be able to apply the $
\backslash
eor$E rule, we'll have to set up two subproofs:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Trabalhar diretmente a partit de uma disjunção funciona um pouco diferente.
 Para usar uma disjunção, usamos a regra 
\begin_inset Formula $\eor$
\end_inset

E.
 A fim de aplicar esta regra, não é suficiente saber quais são os disjuntos
 da disjunção que queremos usar.
 Também devemos ter em mente o que queremos provar.
 Suponha que queremos provar 
\begin_inset space ~
\end_inset


\begin_inset Formula $\meta{C}$
\end_inset

 e temos que trabalhar com 
\begin_inset Formula $\meta{A}\eor B$
\end_inset

 (ou seja, 
\begin_inset Formula $\meta{A}\eor B$
\end_inset

 pode ser uma premissa da prova, uma suposição de uma subprova ou algo já
 foi provado).
 Para conseguirmos aplicar a regra 
\begin_inset Formula $\eor$
\end_inset

E, teremos de estabelecer duas subprovas: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1,(2)-3,(4)-5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The first subproof starts with the first disjunct, $
\backslash
meta{A}$, and ends with the sentence we're looking for, $
\backslash
meta{C}$.
 The second subproof starts with the other disjunct, $
\backslash
meta{B}$, and also ends with the goal sentence~$
\backslash
meta{C}$.
 Each of these subproofs have to be filled in further.
 We can then justify the goal sentence $
\backslash
meta{C}$ by using $
\backslash
eor$E, citing the line with $
\backslash
meta{A} 
\backslash
eor 
\backslash
meta{B}$ and the two subproofs.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A primeira subprova começa com o primeiro disjunto 
\begin_inset Formula $\meta{A}$
\end_inset

 e termina com a sentença que estamos buscando 
\begin_inset Formula $\meta{C}$
\end_inset

.
 A segunda subprova começa com o outro disjunto 
\begin_inset Formula $\meta{B}$
\end_inset

 e também termina com a sentença 
\begin_inset Formula $\meta{C}$
\end_inset

.
 Cada uma destas subprovas tem de ser preenchidas também.
 Podemos, então, justificar a sentença 
\begin_inset Formula $\meta{C}$
\end_inset

, usando-se 
\begin_inset Formula $\eor$
\end_inset

E e citndo-se a linha contendo 
\begin_inset Formula $\meta{A}\eor\meta{B}$
\end_inset

 e as [linhas das] duas subprovas.
\end_layout

\begin_layout Subsection*
Trabalhando diretamente a partir de um condicional
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working forward from a conditional
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%In order to use a conditional $
\backslash
meta{A} 
\backslash
eif 
\backslash
meta{B}$, you also need the antecedent $
\backslash
meta{A}$ in order to apply~$
\backslash
eif$E.
  So to work forward from a conditional, you will derive $
\backslash
meta{B}$, justify it by $
\backslash
eif$E, and set up $
\backslash
meta{A}$ as a new subgoal.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
A fim de usar um condicional 
\begin_inset Formula $\meta{A}\eif\meta{B}$
\end_inset

, você também precisará do antecedente 
\begin_inset Formula $\meta{A}$
\end_inset

 com intuito de aplicar 
\begin_inset Formula $\eif$
\end_inset

E.
 Desse modo, para trabalhar diretamente a partir de um condicional, você
 derivará 
\begin_inset Formula $\meta{B}$
\end_inset

 e justificará esta derivação por meio de 
\begin_inset Formula $\eif$
\end_inset

E e estabelecerá 
\begin_inset Formula $\meta{A}$
\end_inset

 como um novo objetivo secundário.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ce{1,2}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\end_layout

\begin_layout Subsection*
Trabalhando diretamente a partir de uma sentença negada
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working forward from a negated sentence
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Finally, to use a negated sentence $
\backslash
enot 
\backslash
meta{A}$, you would apply $
\backslash
enot$E.
 It requires, in addition to $
\backslash
enot 
\backslash
meta{A}$, also the corresponding sentence~$
\backslash
meta{A}$ without the negation.
 The sentence you'll get is always the same: $
\backslash
ered$.
 So working forward from a negated sentence works especially well inside
 a subproof that you'll want to use for $
\backslash
enot$I (or IP).
  You work forward from $
\backslash
enot 
\backslash
meta{A}$ if you already have $
\backslash
enot 
\backslash
meta{A}$ and you want to prove~$
\backslash
ered$.
 To do it, you set up $
\backslash
meta{A}$ as a new subgoal.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Finalmente, para usar uma sentença negada 
\begin_inset Formula $\enot\meta{A}$
\end_inset

, você deveria aplicar 
\begin_inset Formula $\enot$
\end_inset

E.
 Ela requer, além de 
\begin_inset Formula $\enot\meta{A}$
\end_inset

, também a sentença 
\begin_inset Formula $\meta{A}$
\end_inset

 sem a negação.
 A sentença que você irá obter é sempre a mesma: 
\begin_inset Formula $\ered$
\end_inset

.
 Assim, trabalhar diretamente a partir de uma sentença negada funciona especialm
ente bem dentro de uma subprova na qual você deseja usar 
\begin_inset Formula $\enot$
\end_inset

I (or IP).
 Você trabalha diretamente a partir de 
\begin_inset Formula $\enot\meta{A}$
\end_inset

, se você já tem 
\begin_inset Formula $\enot\meta{A}$
\end_inset

 e deseja provar 
\begin_inset Formula $\ered$
\end_inset

.
 Para fazer isso, você estabelece 
\begin_inset Formula $\meta{A}$
\end_inset

 como novo objetivo secundário.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1,2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Estratégias em funcionamento
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Strategies at work
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Suppose we want to show that the argument $(A 
\backslash
eand B) 
\backslash
eor (A 
\backslash
eand C) 
\backslash
therefore A 
\backslash
eand (B 
\backslash
eor C)$ is valid.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Suponha que desejamos mostrar que o argumento 
\begin_inset Formula $(A\eand B)\eor(A\eand C)\therefore A\eand(B\eor C)$
\end_inset

 é válido.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We start the proof by writing the premise and conclusion down.
 (On a piece of paper, you would want as much space as possible between
 them, so write the premises at the top of the sheet and the conclusion
 at the bottom.)
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Iniciamos a prova escrevendo as premissas e a conclusão (em um pedaço de
 papel, seria desejável ter tanto espaço quanto for possível entre premissas
 e conclusão, portanto escreva as premissas no topo da página e a conlusão
 na parte inferior).
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We now have two options: either work backward from the conclusion, or work
 forward from the premise.
 We'll pick the second strategy: we use the disjunction on line~$1$, and
 set up the subproofs we need for $
\backslash
eor$E.
 The disjunction on line~$1$ has two disjuncts, $A 
\backslash
eand B$ and $A 
\backslash
eand C$.
 The goal sentence you want to prove is $A 
\backslash
eand (B 
\backslash
eor C)$.
 So in this case you have to set up two subproofs, one with assumption $A
 
\backslash
eand B$ and last line $A 
\backslash
eand (B 
\backslash
eor C)$, the other with assumption $A 
\backslash
eand C$ and last line $A 
\backslash
eand (B 
\backslash
eor C)$.
  The justification for the conclusion on line~$n$ will be $
\backslash
eor$E, citing the disjunction on line~$1$ and the two subproofs.
 So your proof now looks like this:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Agora temos duas opções: ou trabalhar de trás para frente a partir da conclusão
 ou trabalhar diretamente a partir das premissas.
 Escolheremos a segunda estratégia: usamos a disjunção na linha 
\begin_inset Formula $1$
\end_inset

, e estabelecemos as subprovas que precisamos para aplicar 
\begin_inset Formula $\eor$
\end_inset

E.
 A disjunção na linha 
\begin_inset Formula $1$
\end_inset

 tem dois disjuntos: 
\begin_inset Formula $A\eand B$
\end_inset

 e 
\begin_inset Formula $A\eand C$
\end_inset

.
 Assim, neste caso, você tem de estabelecer duas subprovas, uma com suposição
 
\begin_inset Formula $A\eand B$
\end_inset

 e com última linha [na subprova] 
\begin_inset Formula $A\eand(B\eor C)$
\end_inset

 e outra subprova com suposição 
\begin_inset Formula $A\eand C$
\end_inset

 e última linha [na subprova] 
\begin_inset Formula $A\eand(B\eor C)$
\end_inset

.
 A justificação para a conclusão será 
\begin_inset Formula $\eor$
\end_inset

E, citando a disjunção na linha 
\begin_inset Formula $1$
\end_inset

 e as duas subprovas.
 Desse modo, sua prova terá a seguinte aparência: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
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\backslash
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\end_inset

 B
\begin_inset ERT
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}
\end_layout

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
have
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\end_inset


\begin_inset ERT
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[
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n
\begin_inset ERT
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\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
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6
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
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\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
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\end_layout

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 C)
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
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\begin_inset ERT
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\backslash
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\begin_inset ERT
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\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
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{
\end_layout

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7
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

11
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

12
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

1,2-6,7-11
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%You now have two separate tasks, namely to fill in each of the two subproofs.
 In the first subproof, we now work backward from the conclusion $A 
\backslash
eand (B 
\backslash
eor C)$.
 That is a conjunction, so inside the first subproof, you will have two
 separate subgoals: proving $A$, and proving $B 
\backslash
eor C$.
 These subgoals will let you justify line~$n$ using~$
\backslash
eand$I.
 Your proof now looks like this:
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Agora temos duas tarefas separadas, a saber, preencher cada uma das duas
 subprovas.
 Na primeira subprova, trabalhamos de trás para frente a partir da conclusão
 
\begin_inset Formula $A\eand(B\eor C)$
\end_inset

.
 Isso é uma conjunção, assim dentro da primeira subprova, você terá dois
 objetivos secundários separados: provar 
\begin_inset Formula $A$
\end_inset

 e provar 
\begin_inset Formula $B\eor C$
\end_inset

.
 Estes objetivos secundários permitem que você justifique a linha 
\begin_inset Formula $n$
\end_inset

 usando 
\begin_inset Formula $\eand$
\end_inset

I.
 Sua prova terá agora a seguinte aparência: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

i
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

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4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

-1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4,5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

11
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

12
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1,2-6,(7)-11
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We immediately see that we can get line $i$ from line~$2$ by $
\backslash
eand$E.
 So line~$i$ is actually line~$3$, and can be justified with $
\backslash
eand$E from line~$2$.
 The other subgoal $B 
\backslash
eor C$ is a disjunction.
 We'll apply the strategy for working backward from a disjunctions to line
 $n-1$.
 We have a choice of which disjunct to pick as a subgoal, $B$ or~$C$.
 Picking $C$ wouldn't work and we'd end up having to backtrack.
 And you can already see that if you pick $B$ as a subgoal, you could get
 that by working forward again from the conjunction $A 
\backslash
eand B$ on line~$2$.
 So we can complete the first subproof as follows:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Imediatamente vemos que podemos obter a linha 
\begin_inset Formula $i$
\end_inset

 a partir da linha 
\begin_inset Formula $2$
\end_inset

 por meio de 
\begin_inset Formula $\eand$
\end_inset

E.
 Assim, a linha 
\begin_inset Formula $i$
\end_inset

 é, de fato, linha 
\begin_inset Formula $3$
\end_inset

 e pode ser justificada por 
\begin_inset Formula $\eand$
\end_inset

E a partir da linha 
\begin_inset Formula $2$
\end_inset

.
 O outro objetivo secundário 
\begin_inset Formula $B\eor C$
\end_inset

 é uma disjunção.
 Aplicaremos a estratégica de trabalhar de trás para frente a partir da
 disjunção na linha 
\begin_inset Formula $n-1$
\end_inset

.
 Temos que escolher qual dos dois disjuntos 
\begin_inset Formula $B$
\end_inset

 ou
\begin_inset space ~
\end_inset


\begin_inset Formula $C$
\end_inset

 selecionamos como objetivo secundário.
 Selecionar 
\begin_inset Formula $C$
\end_inset

 não funcionaria e seríamos obrigados a voltar atrás.
 E você já pode ver que se você selecionar 
\begin_inset Formula $B$
\end_inset

 como objetivo secundário, você poderia obter isso trabalhando diretamente
 a partir da conjunção 
\begin_inset Formula $A\eand B$
\end_inset

 na linha 
\begin_inset Formula $2$
\end_inset

.
 Assim, podemos completar a primeira subprova como se segue: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
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\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3,5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

11
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

12
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1,2-6,7-11
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Like line~$3$, we get line $4$ from $2$ by $
\backslash
eand$E.
 Line~$5$ is justified by $
\backslash
eor$I from line~$4$, since we were working backward from a disjunction there.
  The second subproof is almost exactly the same.
 We'll leave it as an exercise.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Como a linha 
\begin_inset Formula $3$
\end_inset

, obtemos a linha 
\begin_inset Formula $4$
\end_inset

 a partir da linha 
\begin_inset Formula $2$
\end_inset

 por 
\begin_inset Formula $\eand$
\end_inset

E.
 Linha 
\begin_inset Formula $5$
\end_inset

 é justificada por 
\begin_inset Formula $\eor$
\end_inset

I a partir da linha 
\begin_inset Formula $4$
\end_inset

, uma vez que estávamos trabalhando de trás para frente a partir de uma
 disjunção.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%That's it for the first subproof.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Isso é a primeira subprova.
 A segunda subprova é quase exatamente a mesma.
 Ela será deixada como exercício.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Remember that when we started, we had the option of working forward from
 the premise, or working backward from the conclusion, and we picked the
 first option.
 The second option also leads to a proof, but it will look different.
  The first steps would be to work backward from the conclusion and set
 up two subgoals, $A$ and $B 
\backslash
eor C$, and then work forward from the premise to prove them, e.g.,:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Lembre-se de que, quando iniciamos, tínhamos a opção de trabralhar diretamente
 a partir da premissa ou trabalhar de trás para frente a partir da conclusão
 e escolhemos a primeira opção.
 A segunda opção também leva a uma prova, mas ela parecerá diferente.
 Os primeiros passos seriam trabalhar de trás para frente a partir da conclusão
 e estabelecer duas provas secundárias 
\begin_inset Formula $A$
\end_inset

 e 
\begin_inset Formula $B\eor C$
\end_inset

 e, então, trabalhar diretamente das premissas para prová-las.
 Por exemplo: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

-1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1,2-3,(4)-(5)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

9
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

-1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

10
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

11
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1,(7)-8,(9)-(10)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

12
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 (B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6,11
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 Deixaremos para você preencher as partes que estão faltando indicadas por
 
\begin_inset Formula $\vdots$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We'll leave you to fill in the missing pieces indicated by~$
\backslash
vdots$.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Let's give another example to illustrate how to apply the strategies to
 deal with conditionals and negation.
 The sentence $(A 
\backslash
eif B) 
\backslash
eif (
\backslash
enot B 
\backslash
eif 
\backslash
enot A)$ is a tautology.
 Let's see if we can find a proof of it, from no premises, using the strategies.
 We first write the sentence at the bottom of a sheet of paper.
 Since working forward is not an option (there is nothing to work forward
 from), we work backward, and set up a subproof to establish the sentence
 we want $(A 
\backslash
eif B) 
\backslash
eif (
\backslash
enot B 
\backslash
eif 
\backslash
enot A)$ using $
\backslash
eif$I.
 Its assumption must be the antecedent of the conditional we want to prove,
 i.e., $A 
\backslash
eif B$, and its last line the consequent $
\backslash
enot B 
\backslash
eif 
\backslash
enot A$.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Daremos um outro exemplo para ilustrar como aplicar as estratégias para
 lidar com condicionais e negação.
 A sentença 
\begin_inset Formula $(A\eif B)\eif(\enot B\eif\enot A)$
\end_inset

 é uma tautologia.
 Vejamos se podemos encontrar uma prova dela a partir de nenhuma premissa,
 usando as estratégias.
 Primeiro escrevemos a sentença no fim da folha.
 Uma vez que trabalhar diretamente não é uma opção (não há nada para trabalhar
 diretamente), trabalhamos de trás para frente e estipulamos uma subprova
 para estabelecer a sentença que desejamos 
\begin_inset Formula $(A\eif B)\eif(\enot B\eif\enot A)$
\end_inset

, usando 
\begin_inset Formula $\eif$
\end_inset

I.
 A suposição dessa subprova deve ser o antecedente do condicional que desejamos
 provar, ou seja, 
\begin_inset Formula $A\eif B$
\end_inset

 e a última linha da subprova deve ser o consequente de 
\begin_inset Formula $\enot B\eif\enot A$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1-7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 O novo objetivo 
\begin_inset Formula $\enot B\eif\enot A$
\end_inset

 é novamente um condicional, assim trabalhando de trás para frente, estabelecemo
s um outra subprova: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The new goal, $
\backslash
enot B 
\backslash
eif 
\backslash
enot A$ is itself a conditional, so working backward we set up another subproof:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

-1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-(6)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1-7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%From $
\backslash
enot A$ we again work backward.
 To do this, look at the $
\backslash
enot$I rule.
 It requires a subproof with~$A$ as assumption, and $
\backslash
ered$ as its last line.
 So the proof is now:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A partir de 
\begin_inset Formula $\enot A$
\end_inset

, novamente trabalhamos de trás para frente.
 Para fazer isso, olhe para a regra 
\begin_inset Formula $\enot$
\end_inset

I.
 Ela requer um subprova com 
\begin_inset Formula $A$
\end_inset

 como suposição e 
\begin_inset Formula $\ered$
\end_inset

 como a última linha dela.
 Assim, agora a prova é: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

-2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3-(5)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-(6)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1-7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Now our goal is to prove~$
\backslash
ered$.
 We said above, when discussing how to work forward from a negated sentence,
 that the $
\backslash
enot$E rule allows you to prove~$
\backslash
ered$, which is our goal in the innermost subproof.
 So we look for a negated sentence which we can work forward from: that
 would be $
\backslash
enot B$ on line~$2$.
 That means we have to derive $B$ inside the subproof, since $
\backslash
enot$E requires not just $
\backslash
enot B$ (which we have already), but also~$B$.
 And $B$, in turn, we get by working forward from $A 
\backslash
eif B$, since $
\backslash
eif$E will allow us to justify the consequent of that conditional~$B$ by
 $
\backslash
eif$E.
 The rule $
\backslash
eif$E also requires the antecedent~$A$ of the conditional, but that is also
 already available (on line~$3$).
 So we finish with:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Agora nosso objetivo é provar 
\begin_inset Formula $\ered$
\end_inset

.
 Quando discutimos como trabalhar diretamente a partir de uma sentença negada,
 dissemos acima que a regra 
\begin_inset Formula $\enot$
\end_inset

E permite que você prove 
\begin_inset Formula $\ered$
\end_inset

, que é nosso objetivo n subprova mais interna.
 Desse modo, olhamos para um sentença negada com a qual podemos trabalhar
 diretamente: essa seria 
\begin_inset Formula $\enot B$
\end_inset

 na linha 
\begin_inset Formula $2$
\end_inset

.
 Isso significa que temos de drivar 
\begin_inset Formula $B$
\end_inset

 dentro da subprova, uma vez que 
\begin_inset Formula $\enot$
\end_inset

E exige não somente 
\begin_inset Formula $\enot B$
\end_inset

 (que já temos), mas também 
\begin_inset Formula $B$
\end_inset

.
 E 
\begin_inset Formula $B$
\end_inset

 é obtida, por sua vez, trabalhando diretamente a partir de from 
\begin_inset Formula $A\eif B$
\end_inset

, uma vez que 
\begin_inset Formula $\eif$
\end_inset

E permite que justifiquemos o consequente do condicional 
\begin_inset Formula $B$
\end_inset

 por meio de 
\begin_inset Formula $\eif$
\end_inset

E.
 A regra 
\begin_inset Formula $\eif$
\end_inset

E tmbém requer o antecedente 
\begin_inset Formula $A$
\end_inset

 do condicional, mas isso já está também disponível (na linha 
\begin_inset Formula $3$
\end_inset

).
 Assim, terminamos com: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ce{1,3}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2,4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3-5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 B) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 B 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ci
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1-7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Trabalhando diretamente a partir de 
\begin_inset Formula $\ered$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "sec:backred"

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Working forward from $
\backslash
ered$
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%When applying the strategies, you will sometimes find yourself in a situation
 where you can justify~$
\backslash
ered$.
 Using the explosion rule, this would allow you to justify 
\backslash
emph{anything}.
 So $
\backslash
ered$ works like a wildcard in proofs.
 For instance, suppose you want to give a proof of the argument $A 
\backslash
eor B, 
\backslash
enot A 
\backslash
therefore B$.
 You set up your proof, writing the premises $A 
\backslash
eor B$ and $
\backslash
enot A$ at the top on lines $1$ and $2$, and the conclusion~$B$ at the bottom
 of the page.
 $B$ has no main connective, so you can't work backward from it.
 Instead, you must work forward from $A 
\backslash
eor B$: That requires two subproofs, like so:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Ao aplicar as estratégias, às vezes você se encontra em uma situação onde
 você pode justificar 
\begin_inset Formula $\ered$
\end_inset

.
 Usando a regra da explosão, isto permitiria justificar 
\emph on
qualquer coisa
\emph default
.
 Desse modo, 
\begin_inset Formula $\ered$
\end_inset

 funciona como um curinga nas provas.
 Por exemplo, suponha que você deseja dar uma prova do argumento 
\begin_inset Formula $A\eor B,\enot A\therefore B$
\end_inset

.
 Você estabelece a sua prova, escrevendo as premissas 
\begin_inset Formula $A\eor B$
\end_inset

 e 
\begin_inset Formula $\enot A$
\end_inset

 no topo na linhas 
\begin_inset Formula $1$
\end_inset

 e 
\begin_inset Formula $2$
\end_inset

 e a conclusão 
\begin_inset Formula $B$
\end_inset

 na parte inferior da página.
 
\begin_inset Formula $B$
\end_inset

 não tem conectivo principal, assim, você não pode trabalhar de trás para
 frente a partir dele.
 Em vez disso, você deve trabalhar diretamente a partir de 
\begin_inset Formula $A\eor B$
\end_inset

: Isso requer duas subprovas, assim: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1,2-3,(4)-5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Notice that you have $
\backslash
enot A$ on line~$2$ and $A$ as the assumption of your first subproof.
 That gives you $
\backslash
ered$ using $
\backslash
enot$E, and from $
\backslash
ered$ you get the conclusion~$B$ of the first subroof using~X.
 Recall that you can repeat a sentence you already have by using the reiteration
 rule~R.
 So our proof would be:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Observe que você tem 
\begin_inset Formula $\enot A$
\end_inset

 na linha 
\begin_inset Formula $2$
\end_inset

 e 
\begin_inset Formula $A$
\end_inset

 como suposição de nossa primeira subprova.
 Isso lhe dá 
\begin_inset Formula $\ered$
\end_inset

 usando 
\begin_inset Formula $\enot$
\end_inset

E e, a partir de 
\begin_inset Formula $\ered$
\end_inset

, você obtém a conclusão 
\begin_inset Formula $B$
\end_inset

 da primeira subprova usando- X.
 Lembre-se de que você pode repetir uma sentença que você já tem, usando
 a regra de reiteração.
 Desse modo, a prova seria: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7,2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
re
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

1,2-3,4-5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Prosseguindo indiretamente
\end_layout

\begin_layout Standard
Proceed indirectly
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%In very many cases, the strategies of working forward and backward will
 eventually pan out.
 But there are cases where they do not work.
  If you cannot find a way to show $
\backslash
meta{A}$ directly using those, use IP instead.
 To do this, set up a subproof in which you assume $
\backslash
enot
\backslash
meta{A}$ and look for a proof of $
\backslash
ered$ inside that subproof.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Em muitos casos, as estratégias de trabalhar diretamente e de trás para
 frente serão, em geral, bem-sucedidas.
 Mas há casos onde elas não funcionam.
 Se você não consegue achar uma maneira de provar 
\begin_inset Formula $\meta{A}$
\end_inset

 diretamente, use, então, IP.
 Para fazer isto, estabeleça uma subprova na qual você assume 
\begin_inset Formula $\enot\meta{A}$
\end_inset

 e busque uma prova de 
\begin_inset Formula $\ered$
\end_inset

 dentro dessa subprova.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ip
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here, we have to start a subproof with assumption $
\backslash
enot 
\backslash
meta{A}$; the last line of the subproof has to be~$
\backslash
ered$.
 We'll cite the subproof, and use~IP as the rule.
  In the subproof, we now have an additional assumption (on line $n$) to
 work with.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Aqui, temos de iniciar uma subprova com a suposição 
\begin_inset Formula $\enot\meta{A}$
\end_inset

; a última linha da subprova tem de ser 
\begin_inset Formula $\ered$
\end_inset

.
 Citaremos a subprova e usaremos IP como regra.
 Na subprova, temos agora uma suposição adicional (na linha 
\begin_inset Formula $n$
\end_inset

) com a qual podemos trabalhar.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Suppose we used the indirect proof strategy, or we're in some other situation
 where we're looking for a proof of $
\backslash
ered$.
 What's a good candidate? Of course the obvious candidate would be to use
 a negated sentence, since (as we saw above) $
\backslash
enot$E always yields~$
\backslash
ered$.
 If you set up a proof as above, trying to prove 
\backslash
meta{A} using~IP, you will have $
\backslash
enot 
\backslash
meta{A}$ as the assumption of your subproof---so working forward from it
 to justify $
\backslash
ered$ inside your subproof, you would next set up 
\backslash
meta{A} as a goal inside your subproof.
 If you are using this IP strategy, you will find yourself in the following
 situation: 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Suponha que usamos a estratégia de prova indireta ou estamos em alguma outra
 situação onde estamos buscando uma prova de 
\begin_inset Formula $\ered$
\end_inset

.
 Qual é um bom candidato? É claro, o candidato óbvio seria usar um sentença
 negad, uma vez que, como vimos acima, 
\begin_inset Formula $\enot$
\end_inset

E sempre produz
\begin_inset space ~
\end_inset


\begin_inset Formula $\ered$
\end_inset

.
 Se você estabelece uma prova como acima, tentando provar 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 usando
\begin_inset space ~
\end_inset

IP, você terá 
\begin_inset Formula $\enot\meta{A}$
\end_inset

 como suposição de sua subprova — desse modo, trabalhando diretamente dela
 para justificar 
\begin_inset Formula $\ered$
\end_inset

 dentro de sua subprova, você estabeleceria depois 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 como um objetivo dentro da sua subprova.
 Se você está usando esta estratégia IP, você encontrar-se-á na seguinte
 situação: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

-1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2,3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ip
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This looks weird: We wanted to prove $
\backslash
meta{A}$ and the strategies failed us; so we used IP as a last resort.
 And now we find ourselves in the same situation: we are again looking for
 a proof of~$
\backslash
meta{A}$.
 But notice that we are now 
\backslash
emph{inside} a subproof, and in that subproof we have an additional assumption
 ($
\backslash
enot 
\backslash
meta{A}$) to work with which we didn't have before.
 Let's look at an example.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Isto parece estranho: queríamos provar 
\begin_inset Formula $\meta{A}$
\end_inset

 e as estratégias falharam; assim, usamos IP como último recurso.
 E agora nos encontramos na mesma situação: estamos novamente buscando uma
 prova de 
\begin_inset Formula $\meta{A}$
\end_inset

.
 Mas observe que agora estamos 
\emph on
dentro
\emph default
 de uma subprova e, nessa subprova, temos uma suposição adicional (
\begin_inset Formula $\enot\meta{A}$
\end_inset

) com a qual trabalhar que não tínhamos antes.
 Vamos olhar para um exemplo.
\end_layout

\begin_layout Section
Prova indireta do terceiro excluído
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "s:proofLEM"

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Indirect proof of excluded middle
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The sentence $A 
\backslash
eor 
\backslash
enot A$ is a tautology, and so should have a proof even without any premises.
 But working backward fails us: to get $A 
\backslash
eor 
\backslash
enot A$ using $
\backslash
eor$I we would have to prove either $A$ or $
\backslash
enot A$---again, from no premises.
 Neither of these is a tautology, so we won't be able to prove either.
 Working forward doesn't work either, since there is nothing to work forward
 from.
 So, the only option is indirect proof.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

A sentença 
\begin_inset Formula $A\eor\enot A$
\end_inset

 é uma tautologia e, assim, deveríamos ter uma prova mesmo sem premissas.
 Mas trabalhar de trás para frenete falha: para obter 
\begin_inset Formula $A\eor\enot A$
\end_inset

 usando 
\begin_inset Formula $\eor$
\end_inset

I , teríamos de provar 
\begin_inset Formula $A$
\end_inset

 ou 
\begin_inset Formula $\enot A$
\end_inset

 — novamente, de nenhuma premissa.
 Nenhuma delas é uma tautologia, assim não conseguiremos prová-las.
 Trabalhando diretamente não funciona também, pois há nada com que trabalhar
 diretamente.
 Logo, a única opção é prova indireta.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

9
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ip
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Now we do have something to work forward from: the assumption $
\backslash
enot(A 
\backslash
eor 
\backslash
enot A)$.
 To use it, we justify $
\backslash
ered$ by $
\backslash
enot$E, citing the assumption on line~$1$, and also the corresponding unnegated
 sentence $A 
\backslash
eor 
\backslash
enot A$, yet to be proved.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Agora, temos algo com que trabalhar diretamente: a suposição 
\begin_inset Formula $\enot(A\eor\enot A)$
\end_inset

.
 Para usá-la, justificamos 
\begin_inset Formula $\ered$
\end_inset

 por 
\begin_inset Formula $\enot$
\end_inset

E, citando a suposição na linha 
\begin_inset Formula $1$
\end_inset

 e, também, a sentença não-negada 
\begin_inset Formula $A\eor\enot A$
\end_inset

, que ainda será provada.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

-1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2,7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

9
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ip
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%At the outset, working backward to prove $A 
\backslash
eor
\backslash
enot A$ by $
\backslash
eor$I did not work.
 But we are now in a different situation: we want to prove $A 
\backslash
eor
\backslash
enot A$ inside a subproof.
 In general, when dealing with new goals we should go back and start with
 the basic strategies.
 In this case, we should first try to work backward from the disjunction
 $A 
\backslash
eor 
\backslash
enot A$, i.e., we have to pick a disjunct and try to prove it.
 Let's pick~$
\backslash
enot A$.
 This would let us justify $A 
\backslash
eor 
\backslash
enot A$ on line~$m - 1$ using $
\backslash
eor$I.
 Then working backward from $
\backslash
enot A$, we start another subproof in order to justify $
\backslash
enot A$ using $
\backslash
enot$I.
  That subproof must have $A$ as the assumption and~$
\backslash
ered$ as its last line.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
No início, trabalhar de trás para frente para provar 
\begin_inset Formula $A\eor\enot A$
\end_inset

 não funcionou.
 Mas, agora, estamos em uma situação diferente: desejamos provar 
\begin_inset Formula $A\eor\enot A$
\end_inset

 dentro de uma subprova.
 Em geral, ao lidar com novos objetivos, deveríamos voltar atrás e iniciar
 com as estratégias básicas.
 Neste caso, em primeiro lugar, deveríamos tentar trabalhar de trás para
 frente a partir da disjunção 
\begin_inset Formula $A\eor\enot A$
\end_inset

, ou seja, temos de selecionar um disjunto e tentar prová-la [a disjunção].
 Selecionemos 
\begin_inset Formula $\enot A$
\end_inset

.
 Isto permitiria que justificássemos 
\begin_inset Formula $A\eor\enot A$
\end_inset

 na linha 
\begin_inset Formula $m-1$
\end_inset

, usando 
\begin_inset Formula $\eor$
\end_inset

I.
 Então, trabalhando de trás para frente a partir de 
\begin_inset Formula $\enot A$
\end_inset

, iniciamos uma outra subprova a fim de justificar 
\begin_inset Formula $\enot A$
\end_inset

 , usando 
\begin_inset Formula $\enot$
\end_inset

I.
 Essa subprova deve ter 
\begin_inset Formula $A$
\end_inset

 como suposição e 
\begin_inset Formula $\ered$
\end_inset

 como última linha dela.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

-3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3-(5)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2,7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

9
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Inside this new subproof, we again need to justify $
\backslash
ered$.
 The best way to do this is to work forward from a negated sentence; $
\backslash
enot(A 
\backslash
eor 
\backslash
enot A)$ on line~$1$ is the only negated sentence we can use.
 The corresponding unnegated sentence, $A 
\backslash
eor 
\backslash
enot A$, however, directly follows from $A$ (which we have on line~$2$)
 by $
\backslash
eor$I.
 Our complete proof is:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Dentro desta nova subprova, precisamos justificar novamente 
\begin_inset Formula $\ered$
\end_inset

.
 A melhor maneira de se fazer isto é trabalhar diretamente a partir de uma
 sentença negada; 
\begin_inset Formula $\enot(A\eor\enot A)$
\end_inset

 na linha 
\begin_inset Formula $1$
\end_inset

 é a única sentença negada que podemos usar.
 A sentença não-negada correspondente 
\begin_inset Formula $A\eor\enot A$
\end_inset

, todovia, segue-se diretamente de 
\begin_inset Formula $A$
\end_inset

 (a qual temos na linha 
\begin_inset Formula $2$
\end_inset

) por 
\begin_inset Formula $\eor$
\end_inset

I.
 Nossa prova completa é: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2,4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3-5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

6
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2,7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

9
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

2-8
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
practiceproblems
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Use as estratégias para encontrar provas para cada um dos seguintes argumentos:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Use the strategies to find proofs for each of the following arguments:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B,A\eif C\therefore A\eif(B\eand C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eand B)\eif C\therefore A\eif(B\eif C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif(B\eif C)\therefore(A\eif B)\eif(A\eif C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor(B\eand C)\therefore(A\eor B)\eand(A\eor C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eand B)\eor(A\eand C)\therefore A\eand(B\eor C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eor B,A\eif C,B\eif D\therefore C\eor D$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot A\lor\enot B\therefore\enot(A\eand B)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eand\enot B\therefore\enot(A\eif B)$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Formule estratégias trabalhando de trás para frente e diretamente a partir
 de 
\begin_inset Formula $\meta{A}\eiff\meta{B}$
\end_inset

.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Formulate strategies for working backward and forward from $
\backslash
meta{A} 
\backslash
eiff 
\backslash
meta{B}$.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Use as estratégias para encontrar provas para cada uma das seguintes sentenças:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Use the strategies to find proofs for each of the following sentences:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot A\eif(A\eif\ered)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eand\enot A)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $[(A\eif C)\eand(B\eif C)]\eif[(A\lor B)\eif C]$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eif B)\eif(A\eand\enot B)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eor\enot B)\eif(A\eif B)$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 Uma vez que estas deveriam ser provas das sentenças a partir de nenhuma
 premissa, você iniciará com a respectiva sentença na 
\emph on
parte inferior
\emph default
 da prova, que não terá premissas.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Since these should be proofs of sentences from no premises, you will start
 with the respective sentence at the 
\backslash
emph{bottom} of the proof, which will have no premises.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Use as estratégias para encontrar provas para cada um dos seguintes argumentos
 e sentenças:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Use the strategies to find proofs for each one of the following arguments
 and sentences:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot\enot A\eif A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot A\eif\enot B\therefore B\eif A$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif B\therefore\enot A\eor B$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(A\eand B)\eif(\enot A\eor\enot B)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $A\eif(B\eor C)\therefore(A\eif B)\eor(A\eif C)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(A\eif B)\lor(B\eif A)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $((A\eif B)\eif B)\eif B$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset

 Todas estas exigirão a estratégia IP.
 As últimas três são especialmente difíceis.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%These all will require the IP strategy.
 The last three especially are quite hard!
\end_layout

\end_inset


\end_layout

\begin_layout Chapter
Regras adicionais para LVF
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "s:Further"

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Additional rules for TFL
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%In 
\backslash
S
\backslash
ref{s:BasicTFL}, we introduced the basic rules of our proof system for TFL.
 In this section, we will add some additional rules to our system.
 Our extended proof system is a bit easier to work with.
 (However, in 
\backslash
S
\backslash
ref{s:Derived} we will see that they are not strictly speaking 
\backslash
emph{necessary}.)
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicTFL"
plural "false"
caps "false"
noprefix "false"

\end_inset

, introduzimos as regras básicas de nosso sistema prova para LVF.
 Nesta seção, adicionaremos algumas regras adicionais ao nosso sistema.
 Nosso sistema de prova estendido é um pouco mais fácil de se trabalhar
 (todavia, em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:Derived"
plural "false"
caps "false"
noprefix "false"

\end_inset

, veremos que elas não são, estritamente falando, necessárias).
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 
\backslash
section{Reiteration}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% The first additional rule is 
\backslash
emph{reiteration} (R).
 This just allows us to repeat ourselves:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 
\backslash
factoidbox{
\backslash
begin{proof}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
have[m]{a}{
\backslash
meta{A}}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
have[
\backslash
 ]{b}{
\backslash
meta{A}} 
\backslash
by{R}{a}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 
\backslash
end{proof}}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% Such a rule is obviously legitimate; we could have used it in our proof
 in 
\backslash
S
\backslash
ref{sec:backred}:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 
\backslash
begin{proof}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
hypo{1}{A 
\backslash
eor B}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
hypo{7}{
\backslash
enot A}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
open
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
hypo{2}{A} 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
have{8}{
\backslash
ered}
\backslash
ne{7,2} 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
have{3}{B}
\backslash
re{8}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
close 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
open
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
hypo{4}{B}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
have{5}{B}
\backslash
by{R}{4}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
close
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 	
\backslash
have{6}{B}
\backslash
oe{1,2-3,4-5} 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 
\backslash
end{proof}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% This is a fairly typical use of the R rule.
\end_layout

\end_inset


\end_layout

\begin_layout Section
Silogismo disjuntivo
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Disjunctive syllogism
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Aqui está uma forma de argumento muito natural.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here is a very natural argument form.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Quote
Elizabeth está em Massachusetts ou em New York.
 Ela não está em New York.
 Logo, ela está em Massachusetts.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Elizabeth is either in Massachusetts or in DC.
 She is not in DC.
 So, she is in Massachusetts.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
Este padrão de inferência é chamado 
\emph on
silogismo disjuntivo
\emph default
.
 Ela será adicionada ao nosso sistema de prova como se segue: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This inference pattern is called 
\backslash
emph{disjunctive syllogism}.
 We add it to our proof system as follows:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

DS
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab, nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

DS
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab, nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%%As usual, the disjunction and the negation of one disjunct may occur in
 either order and need not be adjacent.
 However, we always cite the disjunction first.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Como de costume, a disjunção e a negação de um dos disjuntos podem ocorrer
 em qualquer ordem e não precisam ser adjacentes.
 Entretanto, sempre citamos a disjunção primeiro.
 
\end_layout

\begin_layout Section
Modus tollens
\end_layout

\begin_layout Standard
\noindent
Um outro padrão útil de inferência é incorporado no seguinte argumento:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Another useful pattern of inference is embodied in the following argument:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Quote
Se Mitt venceu a eleição, então ele está na Casa Branca.
 Ele não está na Casa Branca.
 Logo, ele não venceu a eleição.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%If Mitt has won the election, then he is in the White House.
 He is not in the White House.
 So he has not won the election.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
Este padrão de inferência é chamado 
\emph on
modus tollens
\emph default
.
 A regra correspondente é:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This inference pattern is called 
\backslash
emph{modus tollens}.
 The corresponding rule is:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
mt
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab,a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 Como já dito, as premissas podem ocorrer em qualquer ordem, mas sempre
 citamos o condicional primeiro.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%As usual, the premises may occur in either order, but we always cite the
 conditional first.
 
\end_layout

\end_inset


\end_layout

\begin_layout Section
Eliminação da dupla negação
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Double-negation elimination
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Uma outra regra útil é a 
\emph on
eliminação da dupla negação
\emph default
.
 Esta regra faz exatamente o que ela diz: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Another useful rule is 
\backslash
emph{double-negation elimination}.
 This rule does exactly what it says on the tin:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

dna
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

dna
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 A justificação para isto é que, na linguagem natural, em geral, duplas
 negações cancelam-se.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The justification for this is that, in natural language, double-negations
 tend to cancel out.
 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%That said, you should be aware that context and emphasis can prevent them
 from doing so.
 Consider: `Jane is not 
\backslash
emph{not} happy'.
 Arguably, one cannot infer `Jane is happy', since the first sentence should
 be understood as meaning the same as  `Jane is not 
\backslash
emph{un}happy'.
 This is compatible with `Jane is in a state of profound indifference'.
 As usual, moving to TFL forces us to sacrifice certain nuances of English
 expressions.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Dito isso, você deveria ser consciente que contexto e ênfase podem impedi-las
 de funcionar assim.
 Considere: `Jane não é não feliz'.
 É plausível argumentar que não podemos inferir `Jane é feliz', uma vez
 que a primeira sentença poderia ser entendida como significando o mesmo
 que `Jane não é 
\emph on
in
\emph default
feliz'.
 Isto é compatível com `Jane está em um estado de infiferença profunda'.
 Em geral, ir pata LVF força-nos a sacrificar certas nuances das expressões
 do Português.
 
\end_layout

\begin_layout Section
Terceiro excluído
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Excluded middle
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Suppose that we can show that if it's sunny outside, then Bill will have
 brought an umbrella (for fear of burning).
 Suppose we can also show that, if it's not sunny outside, then Bill will
 have brought an umbrella (for fear of rain).
 Well, there is no third way for the weather to be.
 So, 
\backslash
emph{whatever the weather}, Bill will have brought an umbrella.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Suponha que podemos mostrar que se estiver ensolarado, então Bill levará
 um guarda-chuva (por medo de queimadura).
 Suponha também que podemos mostrar que se não estiver ensolarado, então
 Bill levará guarda-chuva (por medo da chuva).
 Ora, não há terceira via de como estará o tempo.
 Assim, 
\emph on
seja como for o tempo
\emph default
, Bill levará um guarda-chuva.
\end_layout

\begin_layout Standard
Esta linha de pensamento motiva a seguinte regra:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This line of thinking motivates the following rule:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

i
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

j
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

c2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
tnd
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a-c1,b-c2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The rule is sometimes called the law of 
\backslash
emph{excluded middle}, since it encapsulates the idea that 
\backslash
meta{A} may be true or $
\backslash
enot 
\backslash
meta{A}$ may be true, but there is no middle way where neither is true 
\backslash
footnote{You may sometimes find logicians or philosophers talking about
 ``tertium non datur.'' That's the same principle as excluded middle; it
 means ``no third way.'' Logicians who have qualms about indirect proof also
 have qualms about LEM.} Moreover, the subproofs can come in any order, and
 the second subproof does not need to come immediately after the first.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Às vezes, a regra é chamada a lei do 
\emph on
terceiro excluído
\emph default
, uma vez que ela expressa resumidamente a idea segundo a qual 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 pode ser verdadeira ou 
\begin_inset Formula $\enot\meta{A}$
\end_inset

 pode ser verdadeira, mas não há uma terceiro caminho onde nenhum deles
 é verdadeiro
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Às vezes, você pode encontrar lógicos e filósofos falando sobre 
\begin_inset Quotes fls
\end_inset

tertium non datur
\begin_inset Quotes frs
\end_inset

.
 Isso é o mesmo princípio que o terceiro excluído; essa expressão significa
 
\begin_inset Quotes fls
\end_inset

nenhum terceiro caminho
\begin_inset Quotes frs
\end_inset

.
 Lógicos que têm reservas sobre prova indireta também têm reservas sobre
 LEM
\end_layout

\end_inset

.
 Podem existir tantas linhas quantas você quiser entre 
\begin_inset Formula $i$
\end_inset

 e 
\begin_inset Formula $j$
\end_inset

 e tantas linhas quantas você quiser entre 
\begin_inset Formula $k$
\end_inset

 e 
\begin_inset Formula $l$
\end_inset

.
 Além disso, as subprovas podem aparecer em qualquer ordem e a segunda prova
 não precisar aparecer imdiatamente após a primeira.
\end_layout

\begin_layout Standard
Para ver a regra em ação, considere:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%To see the rule in action, consider:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset Formula 
\[
P\therefore(P\eand D)\eor(P\eand\enot D)
\]

\end_inset

Aqui está uma prova que corresponde ao argumento:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here is a proof corresponding with the argument:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a, b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

abo
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 D) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 (P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 D)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

anb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a, nb
\begin_inset ERT
status collapsed

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}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

anbo
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 D) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 (P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 D)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
oi
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

anb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 D) 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 (P 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 D)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
tnd
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b-abo, nb-anbo
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 Aqui está um outro exemplo:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here is another example:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ana
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ce{ana, a}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
tnd
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a-na, na1-na2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\end_layout

\begin_layout Section
Regras de De Morgan
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%De Morgan Rules
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Our final additional rules are called De~Morgan's Laws (named after Augustus
 De~Morgan).
 The shape of the rules should be familiar from truth tables.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Nossas regras adicionais finais são chamadas Leis de De Morgan (devido a
 Augustus De
\begin_inset space ~
\end_inset

Morgan).
 A forma das regras deveria ser familiar a partir das tabelas de verdade.
\end_layout

\begin_layout Standard
A primeira regra de De Morgan é:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The first De Morgan rule is:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

dm
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dem
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 A segunda regra de De Morgan é a inversa da primeira:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The second De Morgan is the reverse of the first:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

dm
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dem
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 A terceira regra de De Morgan é a 
\emph on
dual
\emph default
 da primeira:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The third De Morgan rule is the 
\backslash
emph{dual} of the first:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset


\begin_inset space \space{}
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

dm
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
dem
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 E a quarta é a inversa da terceira:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%And the fourth is the reverse of the third:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

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have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 N
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

red
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 Z
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a4
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

N 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a5
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

N
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a7
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

(N 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

3
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

N 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Dê uma prova para cada um destes argumentos:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Give a proof for each of these arguments:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $E\eor F$
\end_inset

, 
\begin_inset Formula $F\eor G$
\end_inset

, 
\begin_inset Formula $\enot F\therefore E\eand G$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $M\eor(N\eif M)\therefore\enot M\eif\enot N$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(M\eor N)\eand(O\eor P)$
\end_inset

, 
\begin_inset Formula $N\eif P$
\end_inset

, 
\begin_inset Formula $\enot P\therefore M\eand O$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(X\eand Y)\eor(X\eand Z)$
\end_inset

, 
\begin_inset Formula $\enot(X\eand D)$
\end_inset

, 
\begin_inset Formula $D\eor M$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
therefore
\end_layout

\end_inset

 
\begin_inset Formula $M$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Chapter
Conceitos prova-teóricos
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "s:ProofTheoreticConcepts"

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Proof-theoretic concepts
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%In this chapter we will introduce some new vocabulary.
 The following expression: means that there is some proof which starts with
 assumptions among $
\backslash
meta{A}_1, 
\backslash
meta{A}_2, 
\backslash
ldots, 
\backslash
meta{A}_n$ and ends with $
\backslash
meta{C}$ (and contains no undischarged assumptions other than those we started
 with).
 Derivatively, we will write:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Neste capítulo, introduziremos um novo vocabulário.
 A expressão seguinte: 
\begin_inset Formula 
\[
\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}\proves\meta{C}
\]

\end_inset

significa que há alguma prova que inicia com suposições entre 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 e termina com 
\begin_inset Formula $\meta{C}$
\end_inset

 (e não contém quaisquer suposições exceto aquelas com as quais iniciamos)).
 De forma derivada, escreveremos: 
\begin_inset Formula 
\[
\proves\meta{A}
\]

\end_inset

significando que há uma prova de 
\begin_inset Formula $\meta{A}$
\end_inset

 a partir de nenhuma suposição.
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%to mean that there is a proof of $
\backslash
meta{A}$ with no assumptions.
 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The symbol `$
\backslash
proves$' is called the 
\backslash
emph{single turnstile}.
 We want to emphasize that this is not the {double turnstile} symbol (`$
\backslash
entails$') that we introduced in chapter~
\backslash
ref{s:SemanticConcepts} to symbolize entailment.
 The single turnstile, `$
\backslash
proves$', concerns the existence of proofs; the double turnstile, `$
\backslash
entails$', concerns the existence of valuations (or interpretations, when
 used for FOL).
 
\backslash
emph{They are very different notions.}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

O símbolo `
\begin_inset Formula $\proves$
\end_inset

' é chamado 
\emph on
catraca
\emph default
 simples.
 Desejamos enfatizar que isto não é o símbolo da 
\emph on
dupla catraca
\emph default
 (`
\begin_inset Formula $\entails$
\end_inset

') que introduzimos no capítulo
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:SemanticConcepts"
plural "false"
caps "false"
noprefix "false"

\end_inset

 para simbolizar acarretamento.
 A catraca simples, `
\begin_inset Formula $\proves$
\end_inset

', lida com a existência de provas; a dupla catraca, `
\begin_inset Formula $\entails$
\end_inset

', lida com a existência de valorações (ou interpetações, quando usada para
 LPO).
 
\emph on
Elas são noções muito diferentes
\emph default
.
 
\end_layout

\begin_layout Standard
Uma vez que temos nosso símbolo `
\begin_inset Formula $\proves$
\end_inset

', podemos introduzir mais alguma terminologia.
 Para dizer que há uma prova de 
\begin_inset Formula $\meta{A}$
\end_inset

 com nenhuma suposição não-descartada, escrevemos: 
\begin_inset Formula ${}\proves\meta{A}$
\end_inset

.
 Neste caso, dizemos que 
\begin_inset Formula $\meta{A}$
\end_inset

 é um 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

teorema
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset CommandInset label
LatexCommand label
name "def:syntactic_tautology_in_sl"

\end_inset

 
\begin_inset Formula $\meta{A}$
\end_inset

 é um 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

teorema
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 sse 
\begin_inset Formula $\proves\meta{A}$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
Para ilustrar isto, suponha que desejamos mostrar que `
\begin_inset Formula $\enot(A\eand\enot A)$
\end_inset

' é um teorema.
 Desse modo, precisamos de uma prova de `
\begin_inset Formula $\enot(A\eand\enot A)$
\end_inset

', que tem 
\emph on
nenhum
\emph default
 suposição não-descartada.
 Todavia, uma vez que desejamos provar uma sentença cujo operador lógico
 principal é a negação, desejamos iniciar com uma 
\emph on
subprova
\emph default
 dentro da qual assumimos `
\begin_inset Formula $A\eand\enot A$
\end_inset

' e mostramos que esta suposição leva a uma contradição.
 Dito isso, então, a prova é assim:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

red
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na, a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

lnc
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 (A 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 A)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con-red
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%We have therefore proved `$
\backslash
enot (A 
\backslash
eand 
\backslash
enot A)$' on no (undischarged) assumptions.
 This particular theorem is an instance of what is sometimes called 
\backslash
emph{the Law of Non-Contradiction}.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Portanto, provamos `
\begin_inset Formula $\enot(A\eand\enot A)$
\end_inset

' sem usar suposições (não-descartadas).
 Este teorema particular é uma instância do que é chamado, às vezes, 
\emph on
Lei de não-Contradição
\emph default
.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%To show that something is a theorem, you just have to find a suitable proof.
 It is typically much harder to show that something is 
\backslash
emph{not} a theorem.
 To do this, you would have to demonstrate, not just that certain proof
 strategies fail, but that 
\backslash
emph{no} proof is possible.
 Even if you fail in trying to prove a sentence in a thousand different
 ways, perhaps the proof is just too long and complex for you to make out.
 Perhaps you just didn't try hard enough.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Para mostrar que algo é um teorema, você tem de encontrar apenas uma prova
 adequada.
 É tipicamente muito mais difícil mostrar que algo 
\emph on
não
\emph default
 um teorema.
 Para fazer isso, você teria de demonstrar não apenas que certas estratégias
 de prova falham, mas que 
\emph on
nenhuma
\emph default
 prova é possível.
 Mesmo se você falha na tenttiva de provar um sentença em milhares de formas
 diferentes, talvez a prova seja apenas muito longa e complexa para você
 entender.
\end_layout

\begin_layout Standard
Aqui está uma nova terminologia:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here is another new bit of terminology:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 Duas sentenças 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 são 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

dedutivamente equivalentes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 sse cada uma pode ser provada da outra; ou seja, 
\begin_inset Formula $\meta{A}\proves\meta{B}$
\end_inset

 e 
\begin_inset Formula $\meta{B}\proves\meta{A}$
\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Como no caso de mostrar que uma sentença é um teorema, é relativamente fácil
 mostrar que duas sentenças são dedutivamente equivalentes: ela apenas requer
 um par de provas.
 Mostar que sentenças 
\emph on
não
\emph default
 são dedutivamente equivalentes seria muito mais difícil: é tão difícil
 quanto mostrar que uma sentença não é um teorema.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Aqui está uma terceira terminologia relacionada: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
factoidbox
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

 As sentenças 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}$
\end_inset

 são 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

dedutivamente inconsistentes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 sse uma contradição pode ser provada a partir delas, ou seja, 
\begin_inset Formula $\meta{A}_{1},\meta{A}_{2},\ldots,\meta{A}_{n}\proves\ered$
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.
 Se elas não são 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

inconsistentes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

, chamamo-las 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

dedutivamente consistentes
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
É fácil mostrar que algumas sentenças são dedutivamente inconsistentes:
 você precisa apenas provar um contradição a partir da suposição de todas
 as sentenças.
 Mostrar que algumas sentenças não são dedutivamente inconsistentes é muito
 mais difícil.
 Exigiria mais do que apenas fornecer uma prova ou duas; exigiria mostrar
 que nenhuma prova de um certo tipo é 
\emph on
possível
\emph default
.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset Newline newline
\end_inset

 Esta tabela resume se uma ou duas provas são suficientes ou se devemos
 raciocinar todas as provas possíveis.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%This table summarises whether one or two proofs suffice, or whether we
 must reason about all possible proofs.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="3">
<features tabularvalignment="middle">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<column alignment="left" valignment="top">
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
cline{2-3}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\series bold
Sim
\series default
 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\series bold
Não
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%
\backslash
cline{2-3}
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

teorema? 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
uma prova 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" topline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
todas provas possíveis
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
inconsistente? 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
uma prova 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
todas provas possíveis
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
equivalent? 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
duas provas 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
todas provas possíveis
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
consistent? 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
todas provas possíveis 
\end_layout

\end_inset
</cell>
<cell alignment="left" valignment="top" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
uma prova
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
practiceproblems
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Mostre que cada uma das seguintes sentenças é um teorema:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Show that each of the following sentences is a theorem:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $O\eif O$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $N\eor\enot N$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $J\eiff[J\eor(L\eand\enot L)]$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $((A\eif B)\eif A)\eif A$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Apresente provas para mostrar cada um dos seguintes:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Provide proofs to show each of the following:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $C\eif(E\eand G),\enot C\eif G\proves G$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $M\eand(\enot N\eif\enot M)\proves(N\eand M)\eor\enot M$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(Z\eand K)\eiff(Y\eand M),D\eand(D\eif M)\proves Y\eif Z$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $(W\eor X)\eor(Y\eor Z),X\eif Y,\enot Z\proves W\eor Y$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Mostre que cada um dos seguintes pares de sentenças são dedutivamente equivalen
tes:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Show that each of the following pairs of sentences are provably equivalent:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{earg}
\end_layout

\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $R\eiff E$
\end_inset

, 
\begin_inset Formula $E\eiff R$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $G$
\end_inset

, 
\begin_inset Formula $\enot\enot\enot\enot G$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $T\eif S$
\end_inset

, 
\begin_inset Formula $\enot S\eif\enot T$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $U\eif I$
\end_inset

, 
\begin_inset Formula $\enot(U\eand\enot I)$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot(C\eif D),C\eand\enot D$
\end_inset

 
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
item 
\end_layout

\end_inset


\begin_inset Formula $\enot G\eiff H$
\end_inset

, 
\begin_inset Formula $\enot(G\eiff H)$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{earg}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Se você sabe que 
\begin_inset Formula $\meta{A}\proves\meta{B}$
\end_inset

, o que você pode dizer sobre 
\begin_inset Formula $(\meta{A}\eand\meta{C})\proves\meta{B}$
\end_inset

? O que você pode dizer sobre 
\begin_inset Formula $(\meta{A}\eor\meta{C})\proves\meta{B}$
\end_inset

? Explique suas respostas.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%If you know that $
\backslash
meta{A}
\backslash
proves
\backslash
meta{B}$, what can you say about $(
\backslash
meta{A}
\backslash
eand
\backslash
meta{C})
\backslash
proves
\backslash
meta{B}$? What about $(
\backslash
meta{A}
\backslash
eor
\backslash
meta{C})
\backslash
proves
\backslash
meta{B}$? Explain your answers.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash

\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Neste capítulo, reivindicamos que é difícil mostrar que duas sentenças
 não são dedutivamente equivalentes, como também é mostrar que uma sentença
 não é teorema.
 Por que reivindicamos isto? (
\emph on
Dica
\emph default
: pense em um sentença que seria um teorema sse 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 fossem dedutivamente equivalentes).
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%In this chapter, we claimed that it is just as hard to show that two sentences
 are not provably equivalent, as it is to show that a sentence is not a
 theorem.
 Why did we claim this? (
\backslash
emph{Hint}: think of a sentence that would be a theorem iff 
\backslash
meta{A} and 
\backslash
meta{B} were provably equivalent.)
\end_layout

\end_inset


\end_layout

\begin_layout Chapter
Regras derivadas
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "s:Derived"

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Derived rules
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%In this section, we will see why we introduced the rules of our proof system
 in two separate batches.
 In particular, we want to show that the additional rules of 
\backslash
S
\backslash
ref{s:Further} are not strictly speaking necessary, but can be derived from
 the basic rules of 
\backslash
S
\backslash
ref{s:BasicTFL}.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Nesta seção, veremos por que introduzimos as regras de nosso sistema de
 prova em dois lotes separados.
 Em particular, desejamos mostrar que as regras adicionais de 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:Further"
plural "false"
caps "false"
noprefix "false"

\end_inset

 não são, estritamente falando, necessárias, mas podem ser derivadas a partir
 de regras básicas de 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicTFL"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
\end_layout

\begin_layout Section
Derivação de reiteração
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Derivation of Reiteration
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%To illustrate what it means to derive a 
\backslash
emph{rule} from other rules, first consider reiteration.
 It is a basic rule of our system, but it is also not necessary.
 Suppose you have some sentence on some line of your deduction:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Para ilustrar o que significa derivar uma regra de outras regras, considere
 primeiro a reiteração.
 É uma regra básica do sistema, mas também não é necessária.
 Suponha que você tenha alguma sentença em alguma linha da sua dedução:
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%You now want to repeat yourself, on some line $k$.
 You could just invoke the rule~R.
 But equally well, you can do this with other basic rules of 
\backslash
S
\backslash
ref{s:BasicTFL}:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Agora, você deseja repetir 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 em alguma linha 
\begin_inset Formula $k$
\end_inset

.
 Poderíamos apenas invocar a regra
\begin_inset space ~
\end_inset

R.
 Mas, da mesma forma, você pode fazer isto com outras regras básicas de
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicTFL"
plural "false"
caps "false"
noprefix "false"

\end_inset

: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

aa
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ai
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a, a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
ae
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

aa
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%To be clear: this is not a proof.
 Rather, it is a proof 
\backslash
emph{scheme}.
 After all, it uses a variable, `$
\backslash
meta{A}$', rather than a sentence of TFL, but the point is simple: Whatever
 sentences of TFL we plugged in for `$
\backslash
meta{A}$', and whatever lines we were working on, we could produce a bona
 fide proof.
 So you can think of this as a recipe for producing proofs.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Esclarecendo: isto não é uma prova.
 Em vez disso, é um 
\emph on
esquema
\emph default
 de prova.
 Acima de tudo, ela usa uma variável `
\begin_inset Formula $\meta{A}$
\end_inset

', em vez de uma sentença de LVF, mas o ponto é simples: qualquer que seja
 a sentença de LVF que substitua `
\begin_inset Formula $\meta{A}$
\end_inset

' e quaisquer que sejam as linhas que estávamos trabalhando, poderíamos
 produzir uma prova bona fide.
 Assim, podemos pensar nisto como uma receira para produzir provas.
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Indeed, it is a recipe which shows us that, anything we can prove using
 the rule R, we can prove (with one more line) using just the basic rules
 of 
\backslash
S
\backslash
ref{s:BasicTFL} without~R.
 That is what it means to say that the rule~R can be derived from the other
 basic rules: anything that can be justified using~R can be justified using
 only the other basic rules.
\end_layout

\end_inset


\end_layout

\begin_layout Standard
De fato, é uma receita que nos mostra que tudo que podemos provar usando
 a regra R podemos provar (com mais uma linha) usando apenas as regras básicas
 de 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicTFL"
plural "false"
caps "false"
noprefix "false"

\end_inset

 sem
\begin_inset space ~
\end_inset

R.
 Isso é o que significa dizer que a regra R pode ser derivada de outras
 regras básicas: tudo que pode ser justificado usando R pode ser justificado
 usando somente as outras regras básicas.
\end_layout

\begin_layout Section
Derivação do silogismo disjuntivo
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Derivation of Disjunctive Syllogism
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Suponha que você esteja em uma prova e que você tenha algo desta forma:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Suppose that you are in a proof, and you have something of this form:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%You now want, on line $k$, to prove $
\backslash
meta{B}$.
 You can do this with the rule of DS, introduced in 
\backslash
S
\backslash
ref{s:Further}, but equally well, you can do this with the 
\backslash
emph{basic} rules of 
\backslash
S
\backslash
ref{s:BasicTFL}:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Agora, você deseja, na linha 
\begin_inset Formula $k$
\end_inset

, provar 
\begin_inset Formula $\meta{B}$
\end_inset

.
 Você pode fazer isto com a regra de DS, introduzida em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:Further"
plural "false"
caps "false"
noprefix "false"

\end_inset

, mas, da mesma forma, podemos fazer isto com as regras básicas de 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicTFL"
plural "false"
caps "false"
noprefix "false"

\end_inset

:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

red
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na, a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
re
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

red
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
by
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

R
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab, a-b1, b-b2
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%So the DS rule, again, can be derived from our more basic rules.
 Adding it to our system did not make any new proofs possible.
 Anytime you use the DS rule, you could always take a few extra lines and
 prove the same thing using only our basic rules.
 It is a 
\backslash
emph{derived} rule.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Assim, a regra DS pode ser derivada novamente a partir de nossas regras
 mais básicas.
 Adicioná-la a nosso sistema não produzirá qualquer nova prova.
 Toda vez que você usa a regra DS, você sempre poderia tomar algumas linhas
 extras e provar a mesma coisa usando somente nossas regras básicas.
 DS é uma regra 
\emph on
derivada
\emph default
.
\end_layout

\begin_layout Section
Derivação de Modus Tollens
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Derivation of Modus Tollens
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Suponha que você tenha o seguinte na sua prova:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Suppose you have the following in your proof:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%You now want, on line $k$, to prove $
\backslash
enot 
\backslash
meta{A}$.
 You can do this with the rule of MT, introduced in 
\backslash
S
\backslash
ref{s:Further}.
 Equally well, you can do this with the 
\backslash
emph{basic} rules of 
\backslash
S
\backslash
ref{s:BasicTFL}:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Você deseja agora, na linha 
\begin_inset Formula $k$
\end_inset

, provar 
\begin_inset Formula $\enot\meta{A}$
\end_inset

.
 Você pode fazer isto com a regra de MT, introduzida em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:Further"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 Igualmente, você pode fazer isto com as regras 
\emph on
básicas
\emph default
 de 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicTFL"
plural "false"
caps "false"
noprefix "false"

\end_inset

: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

ab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eif
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ce{ab, a}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb, b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

no
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a-nb1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Again, the rule of MT can be derived from the 
\backslash
emph{basic} rules of 
\backslash
S
\backslash
ref{s:BasicTFL}.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Novamente, a regra de MT pode ser derivada a partir das regras básicas de
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicTFL"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 
\end_layout

\begin_layout Section
Derivação da eliminação da dupla negação
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Derivation of Double-Negation Elimination
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Considere o seguinte esquema de dedução:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Consider the following deduction scheme:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

m, a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

con
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ip
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a-a1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Again,  we can derive the DNE rule from the 
\backslash
emph{basic} rules of 
\backslash
S
\backslash
ref{s:BasicTFL}.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Novamente, podemos derivar a regra DNE a partir das regras 
\emph on
básicas
\emph default
 de 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicTFL"
plural "false"
caps "false"
noprefix "false"

\end_inset

.
 
\end_layout

\begin_layout Section
Derivação do terceiro excluído
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Derivation of Excluded Middle
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Suppose you want to prove something using the LEM rule, i.e., you have in
 your proof
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Suponha que você deseja provar algo usando a regra LEM, ou seja, você tem
 em sua prova 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

aaa
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bbb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%You now want, on line $l+1$, to prove $
\backslash
meta{B}$.
 The rule LEM from 
\backslash
S
\backslash
ref{s:Further} would allow you to do it.
 But can do this with the 
\backslash
emph{basic} rules of 
\backslash
S
\backslash
ref{s:BasicTFL}?
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Você deseja agora, na linha 
\begin_inset Formula $l+1$
\end_inset

, provar 
\begin_inset Formula $\meta{B}$
\end_inset

.
 A regra LEM em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:Further"
plural "false"
caps "false"
noprefix "false"

\end_inset

 permitiria que você fizesse isso.
 Mas você pode fazer isto com as regras 
\emph on
básicas
\emph default
 de 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:BasicTFL"
plural "false"
caps "false"
noprefix "false"

\end_inset

?
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%One option is to first prove $
\backslash
meta{A} 
\backslash
eor 
\backslash
enot
\backslash
meta{A}$, and then apply $
\backslash
eor$E, i.e.
\backslash
 proof by cases:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Uma opção é provar primeiro 
\begin_inset Formula $\meta{A}\eor\enot\meta{A}$
\end_inset

, e, então, aplicar 
\begin_inset Formula $\eor$
\end_inset

E, ou seja, prova por casos: 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

aaa
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
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\end_inset

 
\begin_inset ERT
status collapsed

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\backslash
open
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\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

k
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

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b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

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bbb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
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\end_inset

 
\begin_inset ERT
status collapsed

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\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

i
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

tnd
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eor
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

i+1
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

fin
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
oe
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

tnd, a-aaa,b-bbb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%(We gave a proof of $
\backslash
meta{A} 
\backslash
eor 
\backslash
enot
\backslash
meta{A}$ using only our basic rules in 
\backslash
S
\backslash
ref{s:proofLEM}.)
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

(Demos uma prova de 
\begin_inset Formula $\meta{A}\eor\enot\meta{A}$
\end_inset

 usando apenas nossas regras básicas em 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
S
\end_layout

\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "s:proofLEM"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here is another way that is a bit more complicated than the ones before.
 What you have to do is embed your two subproofs inside another subproof.
 The assumption of the subproof will be $
\backslash
enot 
\backslash
meta{B}$, and the last line will be $
\backslash
ered$.
 Thus, the complete subproof is the kind you need to conclude 
\backslash
meta{B} using IP.
 Inside the proof, you'd have to do a bit more work to get~$
\backslash
ered$:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Aqui está outra maneira que é um pouco mais complicada que as anteriores.
 O que você tem de fazer é incorpor (aninhar?) ar duas subprovas dentro
 de uma outra subprova.
 A suposição da subprova será 
\begin_inset Formula $\enot\meta{B}$
\end_inset

 e a última linha será 
\begin_inset Formula $\ered$
\end_inset

.
 Assim, a subprova completa é o tipo que você precisa para concluir 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 usando IP.
 Dentro da prova, você teria de fazer um pouco mais de trabalho para obter
 
\begin_inset Formula $\ered$
\end_inset

:
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

a
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

n
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

aaa
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

aaaa
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb, aaa
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
open
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hypo
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

b
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ellipsesline
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

l
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bbb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bbbb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb, bbb
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

na
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(a)-(aaaa)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nna
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ni
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

(b)-(bbbb)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

bot
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ered
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ne
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nna, na
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
close
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

B
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ip
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nb-(bot)
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
end{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Note that because we add an assumption at the top and additional conclusions
 inside the subproofs, the line numbers change.
 You may have to stare at this for a while before you understand what's
 going on.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Note que, porque adicionamos um suposição no topo e conclusões adicionais
 dentra da subprova, os números de linha mudaram.
 Você pode ter de ficar atento nisto por enquanto, antes que você entenda
 o que está acontecendo.
\end_layout

\begin_layout Section
Derivação das regras de De Morgan
\end_layout

\begin_layout Standard
Aqui está uma demonstração de como poderíamos derivar a primeira regra de
 De Morgan:
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Here is a demonstration of how we could derive the first De Morgan rule:
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
begin{proof}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
have
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

[
\end_layout

\end_inset

m
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

]
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

nab
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
enot
\end_layout

\end_inset

 (
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

A
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

 
\begin_inset ERT
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A
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B
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B
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A
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A
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{
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B
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con
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A
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{
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B
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\backslash
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 Aqui está a demonstração de como poderíamos derivar a segunda regra de
 De Morgan:
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%Here is a demonstration of how we could derive the second De Morgan rule:
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[
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m
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nab
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A
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B
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[
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k
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ab
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a
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B
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na
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nb
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B
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c2
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end{proof}
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%Similar demonstrations can be offered explaining how we could derive the
 third and fourth De Morgan rules.
 These are left as exercises.
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\end_inset

Demonstração similares podem ser oferecidas para explicar como derivar a
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\end_layout

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 Forneça esquemas de prova que justifiquem a adição da terceira e quarta
 regras de De Morgan como regras derivadas.
 
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%Provide proof schemes that justify the addition of the third and fourth
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%The proofs you offered in response to the practice exercises of 
\backslash
S
\backslash
S
\backslash
ref{s:Further}--
\backslash
ref{s:ProofTheoreticConcepts} used derived rules.
 Replace the use of derived rules, in such proofs, with only basic rules.
 You will find some `repetition' in the resulting proofs; in such cases,
 offer a streamlined proof using only basic rules.
  (This will give you a sense, both of the power of derived rules, and of
 how all the rules interact.)
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As provas que você ofereceu em respostas aos exercícios práticos de 
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caps "false"
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–
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 usavam regras derivadas.
 Substitua o uso das regras derivadas em tais provas por apenas regras básicas.
 Você encontrará alguma `repetição' nas provas resultantes; em tais casos,
 ofereça uma prova simplificada usando somente as regras básicas (isto dar-lhe-á
 um sentido do poder das regras derivadas e de como todas as regras interagem).
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 Dê uma prova de 
\begin_inset Formula $\meta{A}\eor\enot\meta{A}$
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.
 Então dê uma prova que 
\emph on
use apenas as regras básicas
\emph default
.
 
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%Give a proof of $
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meta{A} 
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eor 
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enot
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meta{A}$.
 Then
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%give a proof that 
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 Mostre que se você tivesse LEM como uma regra básica, você poderia justificar
 IP como regra derivada.
 Ou seja, suponha que você tivesse a prova: 
\begin_inset ERT
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%Show that if you had LEM as a basic rule, you could justify IP as a derived
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 That is, suppose you had the proof:
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 Como você poderia usá-la para provar 
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A
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 sem usar IP, mas usando LEM assim como todas as regras básicas? 
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%How could you use it to prove 
\backslash
meta{A} without using IP but with using LEM as well as all the other basic
 rules?
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 Dê uma prova da primeira regra de De Morgan, mas usando apenas regras básicas,
 em particular, 
\emph on
sem usar LEM
\emph default
 (é claro, você pode combinar a prova usando LEM com a prova 
\emph on
de
\emph default
 LEM.
 Tente encontrar uma prova direta).
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%Give a proof of the first De Morgan rule, but using only the basic rules,
 in particular, 
\backslash
emph{without using LEM}.
 (Of course, you can combine the proof using LEM with the proof 
\backslash
emph{of}~LEM.
 Try to find a proof directly.)
\end_layout

\end_inset


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\begin_layout Chapter
Corretude e completude
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%Soundness and completeness
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name "sec:soundness_and_completeness"

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%In 
\backslash
S
\backslash
ref{s:ProofTheoreticConcepts}, we saw that we could use derivations to test
 for the same concepts we used truth tables to test for.
 Not only could we use derivations to prove that an argument is valid, we
 could also use them to test if a sentence is a tautology or a pair of sentences
 are equivalent.
 We also started using the single turnstile the same way we used the double
 turnstile.
 If we could prove that 
\backslash
meta{A} was a tautology with a truth table, we wrote $
\backslash
entails 
\backslash
meta{A}$, and if we could prove it using a derivation, we wrote $
\backslash
proves 
\backslash
meta{A}.$ 
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Em 
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S
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, vimos que poderíamos usar derivações para testar os mesmos conceitos que
 testamos usando tabelas de verdade.
 Não somente poderíamos usar derivações para prvar que um argumento é válido,
 também poderíamos usá-las para testar se uma sentença é uma tautologia
 ou se um par de sentenças são equivalentes.
 Também começamos usar a catraca simples da mesma forma que usamos a dupla
 catraca.
 Se pudéssemos provar que 
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A
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 era uma tautologia com tabela de verdade, escreveríamos 
\begin_inset Formula $\entails\meta{A}$
\end_inset

 e se pudéssemos provar isso usando uma derivação, escreveríamos 
\begin_inset Formula $\proves\meta{A}$
\end_inset

.
\end_layout

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%You may have wondered at that point if the two kinds of turnstiles always
 worked the same way.
 If you can show that 
\backslash
meta{A} is a tautology using truth tables, can you also always show that
 it is a theorem using a derivation? Is the reverse true? Are these things
 also true for valid arguments and pairs of equivalent sentences? As it
 turns out, the answer to all these questions and many more like them is
 yes.
 We can show this by defining all these concepts separately and then proving
 them equivalent.
 That is, we imagine that we actually have two notions of validity, valid$_{
\backslash
entails}$ and  valid$_{
\backslash
proves}$ and then show that the two concepts always work the same way.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Você pode ter se perguntado naquele momento se os dois tipos de catracas
 sempre funcionavam da mesma forma.
 Se você pode mostrar que 
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{
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A
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 é uma tautologia usando tabelas de verdade, você pode também sempre mostrar
 que ela é um teorema usando uma derivação? O inverso é verdadeiro? Estas
 coisas são verdadeiras para argumentos e pares de sentenças equivalentes?
 Como se vê, a resposta a todas estas questões e muitas outras semelhantes
 a estas é sim.
 Podemos mostrar isto definindo todos estes conceitos separadamente e, então,
 provando que eles são equivalentes.
 Ou seja, imaginamos que temos, de fato, duas noções de validade — ou seja,
 válido
\begin_inset Formula $_{\entails}$
\end_inset

 e válido
\begin_inset Formula $_{\proves}$
\end_inset

 — e, então, mostramos que os dois conceitos sempre funcionam da mesma forma.
\end_layout

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%To begin with, we need to define all of our logical concepts separately
 for truth tables and derivations.
 A lot of this work has already been done.
 We handled all of the truth table definitions in 
\backslash
S
\backslash
ref{s:SemanticConcepts}.
  We have also already given syntactic definitions for tautologies (theorems)
 and pairs of logically equivalent sentences.
 The other definitions follow naturally.
 For most logical properties we can devise a test using derivations, and
 those that we cannot test for directly can be defined in terms of the concepts
 that we can define.
\end_layout

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\end_layout

\end_inset

Para começar, precisamos definir todos os nossos conceitos lógicos separadamente
 em relação às tabelas de verdade e às derivações.
 Muito deste trabalho já foi feito.
 Lidamos com todas as definições relacionadas às tabelas de verdade em 
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S
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\begin_inset CommandInset ref
LatexCommand ref
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.
 Já demos também definições sintáticas de tautologias (teoremas) e pares
 de sentenças logicamente equivalentes.
 As outras definições seguem-se naturalmente.
 Para muitas propriedades lógicas, podemos elaborar um teste usando derivações
 e aquelas propriedades que não podemos testar diretamente podem ser definidas
 em termos dos conceitos que podemos definir.
\end_layout

\begin_layout Standard
Por exemplo, definimos um teorema como uma sentença que pode ser derivada
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\begin_inset space ~
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\begin_inset CommandInset ref
LatexCommand pageref
reference "def:syntactic_tautology_in_sl"
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).
 Uma vez que a negação de uma contradição é uma tautologia, podemos definir
 uma 
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define
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contradi
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c{c}
\backslash
~a
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o sint
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'a
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tica em LVF
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 como uma sentença cuja negação pode ser derivada sem quaisquer premissas.
 A definição sintática de uma sentença contingentes é um pouco diferente.
 Não temos qualquer método prático, finito para provar que uma sentença
 é contingente usando derivações, a maneira na qual fizemos isso foi usando
 tabelas de verdade.
 Desse modo, temos de nos contentar com a definição de 
\begin_inset Quotes fls
\end_inset

sentença contingente
\begin_inset Quotes frs
\end_inset

 negativamente.
 Uma sentença é 
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sintaticamente contingente em LVF
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\begin_inset CommandInset label
LatexCommand label
name "def:syntactically_contingent_in_sl"

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, se ela não é um teorema nem uma contradição.
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\begin_layout Standard
Uma coleção de sentenças é 
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\end_inset


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{
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dedutivamente inconsistente em LVF
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}
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\begin_inset CommandInset label
LatexCommand label
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 sse podemos derivar uma contradição dessa coleção.
 Consistência, por outro lado, é similar à contingência, já que não temos
 método finito prático para testá-la diretamente.
 Desse modo, novamente, temos de definir um termo negativamente.
 Uma coleção de sentenças é 
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dedutivamente consistente em LVF
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 sse a coleção não é dedutivamente inconsistente.
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\begin_layout Standard
Enfim, uma argumento é 
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{
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dedutivamente v
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\backslash
'a
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lido em LVF
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LatexCommand label
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 sse há uma derivação da conclusão dele a partir das premissas.
 Todas estas definições são dadas na Tabela 
\begin_inset CommandInset ref
LatexCommand ref
reference "table:truth_tables_or_derivations"
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caps "false"
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.
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\backslash
begin{sidewaystable}
\backslash
small 
\backslash
tabulinesep=1ex 
\backslash
begin{tabu}{X[.5,c,m] ||X[1,l,m] |X[1,l,m]} 
\backslash
textbf{Conceito} 		&	
\backslash
textbf{Definição semântica (tabela de verdade)} 	&	
\backslash
textbf{Definição sintática (prova-teórica)} 
\backslash

\backslash
 
\backslash
hline 
\backslash
hline
\end_layout

\begin_layout Plain Layout

Tautologia   &	Uma sentença cuja tabela de verdade tem apenas Vs sob o conectivo
 principal & Uma sentença que pode ser derivada sem quaisquer premissas	
 
\backslash

\backslash
 
\backslash
hline   Contradição		&	Uma sentença cuja tabela de verdade tem apenas Fs
 sob o conectivo principal &	Uma sentença cuja negação pode ser derivada
 sem quaisquer premissas 
\backslash

\backslash
 
\backslash
hline
\end_layout

\begin_layout Plain Layout

Sentença contingente &	Uma sentença cuja tabela de verdade contém tanto
 Vs como Fs sob o conectivo principal & Uma sentença que é nem teorema nem
 contradição 
\backslash

\backslash
 
\backslash
hline
\end_layout

\begin_layout Plain Layout

Sentenças equivalentes &	As colunas sob os conectivos principais são idênticos
 & As sentenças podem ser derivadas uma da outra e vice-versa	
\backslash

\backslash
 
\backslash
hline
\end_layout

\begin_layout Plain Layout

Sentenças insatisfatíveis/inconsistentes &	Sentenças que não têm uma única
 linha nas tabelas de verdade delas onde elas são todas verdadeiras & Sentenças
 a partir das quais se pode derivar uma contradição 
\backslash

\backslash
 
\backslash
hline
\end_layout

\begin_layout Plain Layout

Sentenças satisfatíveis/consistentes	&	Sentenças que têm pelo menos uma
 linha nas suas tabelas de verdade onde elas são todas verdadeiras & Sentenças
 a partir das quais não se pode derivar uma contradição	
\backslash

\backslash
 
\backslash
hline
\end_layout

\begin_layout Plain Layout

Argumento válido		&	Uma argumento cuja tabela de verdade não tem linhas
 onde há Vs sob os conectivos principais das premissas e um F sob o conectivo
 principal da conclusão  & Uma argumento onde podemos derivar a conclusão
 das premissas	
\backslash

\backslash
  
\backslash
end{tabu} 
\backslash
caption{Duas maneiras de se definir conceitos lógicos.} 
\backslash
label{table:truth_tables_or_derivations} 
\backslash
end{sidewaystable}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
Todos nossos conceitos foram agora definidos semantica e sintaticamente.
 Como podemos provar que estas definições sempre funcionam da mesma forma?
 Uma prova completa aqui irá muito além do escopo deste livro.
 Entretanto, podemos esboçar como ela seria.
 Será nosso foco mostrar que as duas noções de validade são equivalentes.
 Desse fato, seguir-se-ão rapidamente os outros conceitos.
 A prova terá de ir em duas direções.
 Em primeiro lugar, teremos de mostrar que coisas que são sintaticamente
 válidas serão também semanticamente válidas.
 Em outras palavras, tudo que podemos provar usando derivações poderia também
 ser provado usando tabelas de verdade.
 Simbolicamente, queremos mostrar que válido
\begin_inset Formula $_{\proves}$
\end_inset

 implica válido
\begin_inset Formula $_{\entails}$
\end_inset

.
 Depois, precisaremos mostrar coisas na outra direção, ou seja, válido
\begin_inset Formula $_{\entails}$
\end_inset

 implica válido
\begin_inset Formula $_{\proves}$
\end_inset

.
\end_layout

\begin_layout Standard
Este argumento de 
\begin_inset Formula $\proves$
\end_inset

 a 
\begin_inset Formula $\entails$
\end_inset

 é o problema da 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

corretude
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "def:soundness"

\end_inset

.
 Um sistema de prova é 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

correto
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

, se não há derivações de argumentos que podem ser mostrados como inválidos
 pelas tabelas de verdade.
\begin_inset CommandInset label
LatexCommand label
name "def_Soundness"

\end_inset

 Demonstrar que o sistema de prova é correto exigiria mostrar que 
\emph on
qualquer
\emph default
 prova possível é a prova de um argumento válido.
 Não seria suficiente simplesmente ser bem-sucedido ao tentar provar muitos
 argumentos válidos e não ser bem-sucedido ao tentar provar argumentos inválidos.
\end_layout

\begin_layout Standard
A prova que esboçaremos depende do fato que nós definimos inicialmente uma
 sentença de LVF, usando uma definição indutiva (veja p.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand pageref
reference "TFLsentences"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
 Poderíamos ter usado definições indutivas para definir uma prova própria
 em LVF e definir uma tabela de verdade própria 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
nix
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

mais tarde, isto será uma atribuição de verdade
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

(todavia, não fizemos assim).
 Se tivássemos estas definições, então poderíamos usar uma 
\emph on
prova indutiva
\emph default
 para mostrar a corretude de LVF.
 Uma prova indutiva funciona da mesma forma que um definição indutiva.
 Com a definição indutiva, identificamos um grupo de elementos básicos que
 são estipulados como sendo exemplos da coisa que estamos tentando definir.
 No caso de uma sentença de LVF, a classe básica era o conjunto de letras
 sentenciais 
\begin_inset Formula $A$
\end_inset

, 
\begin_inset Formula $B$
\end_inset

, 
\begin_inset Formula $C$
\end_inset

, ….
 Anunciamos apenas que estas eram as sentenças.
 O segundo passo de uma definição indutiva é dizer que tudo que é construído
 a partir da sua classe básica usando certas regras também conta como um
 exemplo da coisa que estamos definindo.
 No caso de uma definição de uma sentença, as regras correspondiam aos cinco
 conectivos sentenciais (veja p.
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand pageref
reference "TFLsentences"
plural "false"
caps "false"
noprefix "false"

\end_inset

).
 Uma vez que você estabeleceu uma definição indutiva, você pode usar essa
 definição para mostrar que todos os membros da classe que você definiu
 têm uma certa propriedade.
 Simplesmente, você prova que a propriedade é verdadeira dos membros da
 classe básica e, então, provamos que as regras que estendem a classe básica
 não mudam a propriedade.
 Isto é o que se entende ao dar uma prova indutiva.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Even though we don't have an inductive definition of a proof in TFL, we
 can sketch how an inductive proof of the soundness of TFL would go.
 Imagine a base class of one-line proofs, one for each of our eleven rules
 of inference.
 The members of this class would look like this $
\backslash
meta{A}, 
\backslash
meta{B} 
\backslash
proves  
\backslash
meta{A} 
\backslash
eand 
\backslash
meta{B}$; $
\backslash
meta{A} 
\backslash
eand 
\backslash
meta{B} 
\backslash
proves 
\backslash
meta{A}$; $
\backslash
meta{A} 
\backslash
eor 
\backslash
meta{B}, 
\backslash
enot
\backslash
meta{A} 
\backslash
proves  
\backslash
meta{B}$ 
\backslash
ldots{} etc.
 Since some rules have a couple different forms, we would have to have add
 some members to this base class, for instance $
\backslash
meta{A} 
\backslash
eand 
\backslash
meta{B} 
\backslash
proves  
\backslash
meta{B}$.
  Notice that these are all statements in the metalanguage.
 The proof that TFL is sound is not a part of TFL, because TFL does not
 have the power to talk about itself.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Embora não tenhamos uma definição indutiva de prova em LVF, podemos esboçar
 como uma prova indutiva de corretude de LVF iria.
 Imagine uma classe básica de provas de uma linha, aquela para cada uma
 das onze regras de inferência.
 Os membros desta classe seriam assim: 
\begin_inset Formula $\meta{A},\meta{B}\proves\meta{A}\eand\meta{B}$
\end_inset

; 
\begin_inset Formula $\meta{A}\eand\meta{B}\proves\meta{A}$
\end_inset

; 
\begin_inset Formula $\meta{A}\eor\meta{B},\enot\meta{A}\proves\meta{B}$
\end_inset

 \SpecialChar ldots
 etc.
 Uma vez que algumas regras têm duas formas diferentes, teríamos de ter
 adicionado alguns membros a esta classe básica, por exemplo: 
\begin_inset Formula $\meta{A}\eand\meta{B}\proves\meta{B}$
\end_inset

.
 Observe que estes são todos os enunciados na metalinguagem.
 A prova que LVF é correto não é parte de LVF, porque LVF não tem o poder
 de falar sobre si mesma.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%You can use truth tables to prove to yourself that each of these one-line
 proofs in this base class is valid$_{
\backslash
entails}$.
 For instance the proof $
\backslash
meta{A}, 
\backslash
meta{B} 
\backslash
proves 
\backslash
meta{A} 
\backslash
eand 
\backslash
meta{B}$ corresponds to a truth table that shows $
\backslash
meta{A}, 
\backslash
meta{B} 
\backslash
entails  
\backslash
meta{A} 
\backslash
eand 
\backslash
meta{B}$ This establishes the first part of our inductive proof.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%  
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

Você pode usar tabelas de verdade para provar que cada uma destas provas
 de uma linha nesta classe básica é válida
\begin_inset Formula $_{\entails}$
\end_inset

.
 Por exemplo, a prova 
\begin_inset Formula $\meta{A},\meta{B}\proves\meta{A}\eand\meta{B}$
\end_inset

 corresponde a uma tabela de verdade que mostra 
\begin_inset Formula $\meta{A},\meta{B}\entails\meta{A}\eand\meta{B}$
\end_inset

.
 Isto estabelece a primeira parte de nossa prova indutiva.
\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%The next step is to show that adding lines to any proof will never change
 a valid$_{
\backslash
entails}$ proof into an invalid$_{
\backslash
entails}$ one.
  We would need to do this for each of our eleven basic rules of inference.
 So, for instance, for 
\backslash
eand{I} we need to show that for any proof $
\backslash
meta{A}_{1}$, 
\backslash
dots, $
\backslash
meta{A}_{n} 
\backslash
proves  
\backslash
meta {B}$ adding a line where we use 
\backslash
eand{I} to infer $
\backslash
meta{C} 
\backslash
eand 
\backslash
meta{D}$, where $
\backslash
meta{C} 
\backslash
eand 
\backslash
meta{D}$ can be legitimately inferred from $
\backslash
meta{A}_{1}$, 
\backslash
dots, $
\backslash
meta{A}_{n}$,~$
\backslash
meta{B}$, would not change a valid proof into an invalid proof.
 But wait, if we can legitimately derive $
\backslash
meta{C} 
\backslash
eand 
\backslash
meta{D}$ from these premises, then $
\backslash
meta{C}$ and $
\backslash
meta{D}$ must be already available in the proof.
 They are either already among $
\backslash
meta{A}_{1}$, 
\backslash
dots, $
\backslash
meta{A}_{n}$,~$
\backslash
meta {B}$, or can be legitimately derived from them.
  As such, any truth table line in which the premises are true must be a
 truth table line in which 
\backslash
meta{C} and 
\backslash
meta{D} are true.
   According to the characteristic truth table for 
\backslash
eand, this means that $
\backslash
meta{C} 
\backslash
eand 
\backslash
meta{D}$ is also true on that line.
 Therefore, $
\backslash
meta{C} 
\backslash
eand 
\backslash
meta{D}$ validly follows from the premises.
 This means that using the {
\backslash
eand}E rule to extend a valid proof produces another valid proof.
 This tedious exercise falls beyond the scope of this book.
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

% 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset

O próximo passo é mostrar que adicionar linhas a qualquer prova nunca mudará
 uma prova válida
\begin_inset Formula $_{\entails}$
\end_inset

 para uma prova inválida
\begin_inset Formula $_{\entails}$
\end_inset

.
 Precisaríamos fazer isto para cada um das nossas onze regras de inferência.
 Assim, por exemplo, para 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

I
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 precisamos mostrar que, para qualquer 
\begin_inset Formula $\meta{A}_{1}$
\end_inset

, …, 
\begin_inset Formula $\meta{A}_{n}\proves\meta{B}$
\end_inset

, adicionar uma linha onde usamos 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

I
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 para inferir 
\begin_inset Formula $\meta{C}\eand\meta{D}$
\end_inset

 — onde 
\begin_inset Formula $\meta{C}\eand\meta{D}$
\end_inset

 pode ser legitimamente inferida de 
\begin_inset Formula $\meta{A}_{1}$
\end_inset

, …, 
\begin_inset Formula $\meta{A}_{n}$
\end_inset

,
\begin_inset space ~
\end_inset


\begin_inset Formula $\meta{B}$
\end_inset

 — não mudaria uma prova válida para uma prova inválida.
 Mas espere, se podemos legitimamente derivar 
\begin_inset Formula $\meta{C}\eand\meta{D}$
\end_inset

 destas premissas, então 
\begin_inset Formula $\meta{C}$
\end_inset

 and 
\begin_inset Formula $\meta{D}$
\end_inset

 devem já estar disponíveis na prova.
 Elas já estão entre 
\begin_inset Formula $\meta{A}_{1}$
\end_inset

, …, 
\begin_inset Formula $\meta{A}_{n}$
\end_inset

,
\begin_inset space ~
\end_inset


\begin_inset Formula $\meta{B}$
\end_inset

 ou podem ser ligitimamente derivadas a partir de 
\begin_inset Formula $\meta{A}_{1}$
\end_inset

, …, 
\begin_inset Formula $\meta{A}_{n}$
\end_inset

,
\begin_inset space ~
\end_inset


\begin_inset Formula $\meta{B}$
\end_inset

.
 Como tal, qualquer linha da tabela de verdade na qual as premissas são
 verdadeiras deve ser uma linha da tabela de verdade na qual 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

C
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 e 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
meta
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

D
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 são verdadeiras.
 De acordo com a tabela de verdade característica para 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset

, isto significa que 
\begin_inset Formula $\meta{C}\eand\meta{D}$
\end_inset

 é também verdadeira nesta linha.
 Portanto, 
\begin_inset Formula $\meta{C}\eand\meta{D}$
\end_inset

 segue-se validamente das premissas.
 Isto significa que usar a regra 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
eand
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

E para estender uma prova válida produz uma outra prova válida.
\end_layout

\begin_layout Standard
A fim de mostrar que o sistema de prova é correto, precisaríamos mostrar
 isto para as outras regras de inferência.
 Uma vez que as regras derivadas são consequências das regras básicas, seria
 suficiente fornecer argumentos similares para as outras 11 regras básicas.
 Este exercício tedioso está além do escopo deste livro.
\end_layout

\begin_layout Standard
Desse modo, mostramos que 
\begin_inset Formula $\meta{A}\proves\meta{B}$
\end_inset

 implies 
\begin_inset Formula $\meta{A}\entails\meta{B}$
\end_inset

.
 A outra direção diz que 
\emph on
todo
\emph default
 argumento que pode ser mostrado válido usando tabelas de verdade pode ser
 provado usando uma derivação.
\end_layout

\begin_layout Standard
Este é o problema da completude.
 Um sistema de prova tem a propriedade de 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
define
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

completude
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

 
\begin_inset CommandInset label
LatexCommand label
name "def:completeness"

\end_inset

 sse há uma derivação de qualquer argumento semanticamente válido.
 Provar que um sistema é completo é, em geral, mais difícil que provar que
 é é correto.
 Provar que um sistema é correto equivale a mostrar que todas as regras
 de nosso sistema de prova funcionam como elas deveriam supostamente funcionar.
 Mostrar que um sistema é completo significa mostrar que você incluiu 
\emph on
todas
\emph default
 as regras que você precisa, que você não deixou qualquer coisa de fora.
 Mostrar isto está além do escopo deste livro.
 O ponto importante é que, felizmente, o sistema de prova para LVF é tanto
 correto como completo.
 Isto não é o caso para todos os sistemas de prova ou de todas linguagens
 formais.
 Porque isso é verdadeiro de LVF, podemos escolher dar provas ou dar tabelas
 de verdade — o que for mais fácil para tarefa em questão.
\end_layout

\begin_layout Standard
Agora que sabemos que o método de tabela de verdade é intercambiável com
 o método de derivações, você pode escolher qual método você deseja usar
 para resolver qualquer problema dado.
 Frequentemente, estudantes preferem usar tabelas de verdade, porque elas
 podem ser produzidas de forma puramente mecânica e que parece ser `mais
 fácil'.
 Entretanto, já vimos que tabelas de verdade tornam-se inviavelmente largas
 com algumas poucas letras sentenciais.
 Por outro lado, há duas situações onde usar provas simplesmente não é possível.
 Definimos sintaticamente uma sentença contingente como uma sentença que
 não poderia ser provada como sendo uma tautologia ou uma contradição.
 Não há maneira prática para provar este tipo de enunciado negativo.
 Nunca saberemos se não há alguma prova lá fora que um enunciado é uma contradiç
ão e só não a encontramos ainda.
 Não há nada a fazer nessa situação, exceto recorrer às tabelas de verdade.
 Da mesma forma, podemos usar derivações para provar que duas sentenças
 são equivalentes, mas o que podemos dizer se desejamos provar que elas
 não são equivalentes? Não temos nenhuma maneira de provar que nunca encontrarem
os a prova relevante.
 Desse modo, temos de retornar às tabelas de verdade.
\end_layout

\begin_layout Standard
A tabela 
\begin_inset CommandInset ref
LatexCommand ref
reference "table.ProofOrModel"
plural "false"
caps "false"
noprefix "false"

\end_inset

 resume quando é melhor dar provas e uando é melhor dar tabelas de verdade.
 
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
begin{table}
\backslash
small 
\backslash
tabulinesep=1ex 
\end_layout

\begin_layout Plain Layout


\backslash
begin{tabu}{X[.7,c,m] ||X[1,l,m] |X[1,l,m]} 
\end_layout

\begin_layout Plain Layout


\backslash
textbf{Propriedade lógica} 	&	
\backslash
textbf{Provar que a propriedade está presente} 	&	
\backslash
textbf{Provar que a propriedade está faltando} 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hline 
\backslash
hline 
\end_layout

\begin_layout Plain Layout

Ser um teorema 		& Derive a sentença  						& Encontre uma linha falsa na
 tabela de verdade para a sentença 
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hline 
\end_layout

\begin_layout Plain Layout

Ser uma contradição	&  Derive a negação da sentença  		 & Encontre uma linha
 verdadeira na tabela de verdade para a sentença
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hline 
\end_layout

\begin_layout Plain Layout

Contingência 			& Encontre uma linha falsa e uma linha verdadeira na tabela
 de verdade para a sentença & Prove a sentença ou a negação dela
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hline 
\end_layout

\begin_layout Plain Layout

Equivalência 			& Derive cada sentença da outra e vice-versa 		 & Encontre
 uma linha nas tabelas de verdade para as sentenças onde elas têm valores
 diferentes
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hline 
\end_layout

\begin_layout Plain Layout

Consistência 		& Encontre uma linha na tabela de verdade para as sentenças
 onde elas são todas verdadeiras & Derive uma contradição das sentenças
\backslash

\backslash

\end_layout

\begin_layout Plain Layout


\backslash
hline 
\end_layout

\begin_layout Plain Layout

Validade 				& Derive a conclusão das premissas & Encontre uma linha na
 tabela de verdade onde as premissas são verdadeiras e a conclusão é falsa.
 
\backslash

\backslash
  
\end_layout

\begin_layout Plain Layout


\backslash
end{tabu} 
\end_layout

\begin_layout Plain Layout


\backslash
caption{Quando usar uma tabela de verdade e quando dar uma prova.} 
\end_layout

\begin_layout Plain Layout


\backslash
label{table.ProofOrModel}
\end_layout

\begin_layout Plain Layout


\backslash
end{table}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
practiceproblems
\end_layout

\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Use uma derivação ou uma tabela de verdade para cada um dos seguintes:
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Use either a derivation or a truth table for each of the following.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%[label=(
\backslash
arabic*)]
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $A\eif[((B\eand C)\eor D)\eif A]$
\end_inset

 é um teorema.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $A\eif(A\eif B)$
\end_inset

 não é um teorema.
 
\end_layout

\begin_layout Enumerate
Mostre que a sentença 
\begin_inset Formula $A\eif\enot{A}$
\end_inset

 não é uma contradição.
 
\end_layout

\begin_layout Enumerate
Mostre que a sentença 
\begin_inset Formula $A\eiff\enot A$
\end_inset

 é uma contradição.
 
\end_layout

\begin_layout Enumerate
Mostre que a sentença 
\begin_inset Formula $\enot(W\eif(J\eor J))$
\end_inset

 é contingente.
 
\end_layout

\begin_layout Enumerate
Mostre que a sentença 
\begin_inset Formula $\enot(X\eor(Y\eor Z))\eor(X\eor(Y\eor Z))$
\end_inset

 não é contingente.
 
\end_layout

\begin_layout Enumerate
Mostre que a sentença 
\begin_inset Formula $B\eif\enot S$
\end_inset

 é equivalente à sentença 
\begin_inset Formula $\enot\enot B\eif\enot S$
\end_inset

.
 
\end_layout

\begin_layout Enumerate
Mostre que a sentença 
\begin_inset Formula $\enot(X\eor O)$
\end_inset

 não é equivalente à sentença 
\begin_inset Formula $X\eand O$
\end_inset

.
 
\end_layout

\begin_layout Enumerate
Mostre que as sentenças 
\begin_inset Formula $\enot(A\eor B)$
\end_inset

, 
\begin_inset Formula $C$
\end_inset

, 
\begin_inset Formula $C\eif A$
\end_inset

 são conjuntamente inconsistentes.
 
\end_layout

\begin_layout Enumerate
Mostre que as sentenças 
\begin_inset Formula $\enot(A\eor B)$
\end_inset

, 
\begin_inset Formula $\enot{B}$
\end_inset

, 
\begin_inset Formula $B\eif A$
\end_inset

 são conjuntamente consistentes.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $\enot(A\eor(B\eor C))$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
therefore
\end_layout

\end_inset

 
\begin_inset Formula $\enot{C}$
\end_inset

 é válido.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $\enot(A\eand(B\eor C))$
\end_inset

 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
therefore
\end_layout

\end_inset

 
\begin_inset Formula $\enot{C}$
\end_inset

 é inválido.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
problempart
\end_layout

\end_inset

 Use uma derivação ou uma tabela de verdade para cada um dos seguintes:
 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%Use either a derivation or a truth table for each of the following.
 
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

%[label=(
\backslash
arabic*)]
\end_layout

\begin_layout Plain Layout

\end_layout

\end_inset


\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $A\eif(B\eif A)$
\end_inset

 é um teorema.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $\enot(((N\eiff Q)\eor Q)\eor N)$
\end_inset

 não é um teorema.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $Z\eor(\enot Z\eiff Z)$
\end_inset

 é contingente.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $(L\eiff((N\eif N)\eif L))\eor H$
\end_inset

 não é contingente.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $(A\eiff A)\eand(B\eand\enot B)$
\end_inset

 é uma contradição.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $(B\eiff(C\eor B))$
\end_inset

 não é uma contradição.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $((\enot X\eiff X)\eor X)$
\end_inset

 é equivalente a 
\begin_inset Formula $X$
\end_inset

.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $F\eand(K\eand R)$
\end_inset

 não é equivalente a 
\begin_inset Formula $(F\eiff(K\eiff R))$
\end_inset

.
 
\end_layout

\begin_layout Enumerate
Mostre que as sentenças 
\begin_inset Formula $\enot(W\eif W)$
\end_inset

, 
\begin_inset Formula $(W\eiff W)\eand W$
\end_inset

, 
\begin_inset Formula $E\eor(W\eif\enot(E\eand W))$
\end_inset

 não conjuntamente inconsistentes.
 
\end_layout

\begin_layout Enumerate
Mostre que as sentenças 
\begin_inset Formula $\enot R\eor C$
\end_inset

, 
\begin_inset Formula $(C\eand R)\eif\enot R$
\end_inset

, 
\begin_inset Formula $(\enot(R\eor R)\eif R)$
\end_inset

 são conjuntamente consistentes.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $\enot\enot(C\eiff\enot C),((G\eor C)\eor G)\therefore((G\eif C)\eand G)$
\end_inset

 é válido.
 
\end_layout

\begin_layout Enumerate
Mostre que 
\begin_inset Formula $\enot\enot L,(C\eif\enot L)\eif C)\therefore\enot C$
\end_inset

 é inválido.
 
\end_layout

\end_body
\end_document
